On an initial inverse problem for a diffusion equation with a conformable derivative

Abstract

In this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

1 Introduction

In this paper we consider the following diffusion equation:

$$ \textrm {D}_{\upgamma }^{0}\mathfrak {u}(\textrm {t},\textrm {x}) - \mathfrak {B}\mathfrak {u}(\textrm {t},\textrm {x}) = \textrm {F}(\textrm {t},\textrm {x}), \quad (\textrm {t},\textrm {x}) \in (0,\mathscr{T}_{o}) \times \boldsymbol {\varOmega }, $$

(1)

subject to the boundary conditions

$$\begin{aligned} \mathfrak {u}(\textrm {t},\textrm {x}) = 0 , \quad (\textrm {t},\textrm {x}) \in (0,\mathscr{T}_{o}) \times \partial \boldsymbol {\varOmega }, \end{aligned}$$

(2)

and the initial condition

$$\begin{aligned} \mathfrak {u}(0,\textrm {x}) = \mathfrak {u}_{0}(\textrm {x}) , \quad \textrm {x}\in {\boldsymbol {\varOmega }} \end{aligned}$$

(3)

where the domain Ω is a subset of a d-dimensional space $\mathbb{R}^{d}$ ($d=1,2,3$ is the dimension of Ω), which is a bounded domain with sufficient smooth boundary ∂Ω, F is the source term, $\mathscr{T}_{o}> 0$ is a fixed value, and $\textrm {D}_{\upgamma }^{0}$ is the conformable derivative [4, 14, 16].

Fractional differential equations are successful models of real life phenomenon and many authors studied fractional partial differential equations (see e.g. [34,35,36]). This gives one motivation to study and discuss some of the well known classical differential equations, when some classical derivatives are replaced by fractional derivatives. One of the classical equations is the diffusion equation with the conformable derivative and because of the relationship between the conformable derivative and the classical derivative our equation could be considered as a modified classical diffusion equation.

Khalid et al. [16] introduced the conformable derivative and it was developed in [2,3,4, 6,7,8,9, 31, 33]. Applications of this derivative were given in [5, 14, 25, 32]. In [22] the existence of solutions to conformable nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term was discussed and in [29] the authors presented the iterative learning control for conformable differential equations. Recently, Machado et al. [1] and Baleanu et al. [15, 30] mentioned the critical analysis of the conformable derivative.

If the initial data $\mathfrak {u}_{0}$ and the source term F are given, Problem (1) satisfying (2) and (3) is called the direct problem. The inverse problem for (1) is less well known. Inverse problems occur when we do not know all the given data. However, by adding some given data, we can discuss inverse problems such as the backward problem (recovering the initial data) or the source identification problem (recovering the source function). Initial inverse problems for fractional Riemann–Liouville or Caputo diffusion equations were discussed in the literature [20, 26, 27]. However, little is known on the initial inverse problem for the diffusion equation with a conformable derivative.

Motivated by the above, in this paper, we study the initial inverse problem of the diffusion equation with a conformable derivative (1) satisfying (2) and we reconstruct the initial data $\mathfrak {g}(\textrm {x})=\mathfrak {u}(0,\textrm {x})$ from the additional data

$$ \mathfrak {u}(\mathscr{T}_{o},\textrm {x}) = \textrm {h}(\textrm {x}) , \quad \textrm {x}\in {\boldsymbol {\varOmega }}. $$

(4)

Note we cannot observe the data $(\textrm {h}, \textrm {F})$, so we only get approximate data $(\textrm {h}^{\upepsilon }, \textrm {F}^{\upepsilon })$ such that

$$\begin{aligned} \bigl\Vert \textrm {h}- \textrm {h}^{\upepsilon }\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert {\textrm {F}- \textrm {F}^{\upepsilon }} \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \le \upepsilon , \end{aligned}$$

(5)

where $\upepsilon > 0$ is the noise level (in this paper we will also let $\| \cdot\|$ denote the $L^{2}(\boldsymbol {\varOmega })$ norm).

We show that this problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. Indeed, a small error of the given observation can result in that the solution may have a large error. Some regularization method is required for constructing stable approximations for a sought solution.

We use the Landweber method to find a regularized solution and the idea is based on iterative sequences. Using this method, some authors established a fractional method for solving some linear ill-posed models; see, for example, [13, 23]. We will consider regularized solutions and regularity for the regularized solution. Also, we present an error estimate of the Landweber regularized solution to the exact solution under an a priori assumption using an a priori regularization parameter choice rule, which depends on the noise level ε and the a priori bound condition E of the unknown solution. That means the a priori choice of the regularization parameter depends on the a priori bound of the unknown solution. However, an a priori bound cannot be known exactly in practice, and working with an incorrect value may lead to a bad regularized solution. Therefore, we provide an a posteriori choice of the regularization parameter. We also present a regularized problem and consider the well-posedness of the regularized solution and an error estimate under two parameter choice rules are considered.

The structure of this paper is as follows. First, we give some preliminaries which are needed for this paper in Sect. 2. Next, in Sect. 3, we construct an approximate regularized solution by using the Landweber regularization method. Finally, we estimate the error between the approximation and the sought solution under two parameter choice rules in Sect. 4.

2 Preliminaries

2.1 Some basic results

In this section, we introduce some spaces and some basic definitions associated with conformable derivatives.

Definition 2.1

(Conformable Derivative)

Let $\textrm {f}: [0,\infty ) \to \mathbb{R}$. Then

$$\begin{aligned} \textrm {D}_{\upgamma }^{0}\textrm {f}(\textrm {s})=\lim_{\upepsilon \to 0} \frac{\textrm {f}(\textrm {s}+\upepsilon \textrm {s}^{1-\upgamma }) - \textrm {f}(\textrm {s})}{\upepsilon }, \quad \textrm {s}>0, \end{aligned}$$

(6)

is called the conformable derivative of f of order $\upgamma \in (0,1]$.

Some properties of the conformable derivative can be found in [2, 4, 14] and the references therein.

Consider the operator $-\mathfrak {B}$ on $L^{2}(\boldsymbol {\varOmega })$ with domain $D(-\mathfrak {B}) \subset H_{0}^{1}(\boldsymbol {\varOmega }) \cap H^{2}(\boldsymbol {\varOmega })$. Assume that $-\mathfrak {B}$ has eigenvalues $\{\widetilde{\textrm{a}}_{\textrm {m}}\}$ satisfying

$$ 0 < \widetilde{\textrm{a}}_{1} \le \widetilde{\textrm{a}}_{2} \le \widetilde{\textrm{a}}_{3} \le \cdots \le \widetilde{ \textrm{a}}_{\textrm {m}} \le \cdots $$

and $\widetilde{\textrm{a}}_{\textrm {m}}\to \infty$ as $\textrm {m}\to \infty$ with corresponding eigenfunctions $\textrm{e}_{\textrm {m}}\in H_{0}^{1}(\boldsymbol {\varOmega }) \cap H^{2}(\boldsymbol {\varOmega })$. Now

$$ \textstyle\begin{cases} \mathfrak {B}\textrm{e}_{\textrm {m}}(\textrm {x}) = -\widetilde{\textrm{a}}_{\textrm {m}}\textrm{e} _{\textrm {m}}(\textrm {x}), & \textrm {x}\in \boldsymbol {\varOmega }, \\ \textrm{e}_{\textrm {m}}(\textrm {x}) = 0, & \textrm {x}\in \partial \boldsymbol {\varOmega }, \end{cases} $$

and we note that there exists a positive constant C such that $\widetilde{\textrm{a}}_{\textrm {m}}\ge C \textrm {m}^{\frac{2}{d}}$ for $\textrm {m}\in \mathbb{N}$ and $\textrm {m}\ge 1$, where d is the dimension of the domain Ω; see [10].

For $\textrm {r}\ge 0$, consider the Hilbert scale space (see [21])

$$\begin{aligned} \mathcal{H}^{\textrm {r}}(\boldsymbol {\varOmega })= \Biggl\{ v \in L^{2}(\varOmega ) : \sum_{\textrm {k}=1} ^{\infty } \widetilde{ \textrm{a}}_{\textrm {m}}^{\textrm {r}} \bigl\vert \langle v, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} < +\infty \Biggr\} , \end{aligned}$$

(7)

with the norm

$$\begin{aligned} \Vert v \Vert _{\mathcal{H}^{\textrm {r}}(\boldsymbol {\varOmega })} = \Biggl( \sum _{\textrm {k}=1}^{\infty } \widetilde{\textrm{a}}_{\textrm {m}}^{\textrm {r}} \bigl\vert \langle v,\textrm{e} _{\textrm {m}} \rangle \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. \end{aligned}$$

If $\textrm {r}=0$, we have $\mathcal{H}^{0}(\boldsymbol {\varOmega }) = L^{2}(\boldsymbol {\varOmega })$.

For a given real number $\textrm {p}\ge 1$, let $L^{\textrm {p}}(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$ be the space of all functions such that

$$\begin{aligned} \Vert v \Vert _{L^{\textrm {p}}(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} := \biggl( \int _{0}^{\mathscr{T} _{o}} \bigl\Vert v(\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{\textrm {p}}\,d\textrm {t}\biggr)^{\frac{1}{\textrm {p}}} < + \infty . \end{aligned}$$

We give two lemmas, which will be needed later.

Lemma 2.1

([19, 28])

For $0 < \uplambda < 1$, $\textrm {r}> 0$and $\textrm {k}\in \mathbb{N}$, we have

$$\begin{aligned} (1 - \uplambda )^{\textrm {k}} \uplambda ^{\textrm {r}} \le \textrm {r}^{\textrm {r}} (\textrm {k}+ 1) ^{-\textrm {r}} < \textrm {r}^{\textrm {r}} \textrm {k}^{-\textrm {r}}. \end{aligned}$$

Lemma 2.2

For $\widetilde{\textrm{a}}_{\textrm {m}}> 0$, $\upgamma > 0$, $\upalpha \in ( \frac{1}{2},1]$and $0 < \upmu \exp ( -2\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } ) < 1$, we get

$$\begin{aligned} \sup_{\widetilde{\textrm{a}}_{\textrm {m}}>0} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}}. \end{aligned}$$

Proof

First, we obtain

$$\begin{aligned} &\exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ 1 - \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o} ^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \\ &\quad = \upmu ^{\frac{1}{2}} \biggl[ \upmu ^{\frac{1}{2}} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr]^{-1} \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } . \end{aligned}$$

(8)

Let

$$\begin{aligned} \varPsi (\upnu ) := \upnu ^{-2} \bigl[ 1- \bigl( 1- \upnu ^{2} \bigr)^{\textrm {k}} \bigr]^{2\upalpha }, \end{aligned}$$

where $\upmu := \upmu ^{\frac{1}{2}} \exp ( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } )$. We note that $0 < \upmu < \frac{1}{\| \mathcal {K}\|^{2}}$ (see [18]), and this implies that $0 < \upmu \exp ( - 2 \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } ) < 1$. Hence, this function is continuous in $[0, +\infty )$ when $\upmu \in (0,1)$.

For $\upalpha \in (\frac{1}{2}, 1)$ and $\upmu \in (0,1)$, from Lemma 3.3 in [18], we obtain

$$\begin{aligned} \varPsi (\upnu ) \le \textrm {k}. \end{aligned}$$

(9)

Combining (8) and (9), we deduce that

$$\begin{aligned} \sup_{\widetilde{\textrm{a}}_{\textrm {m}}>0} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}}. \end{aligned}$$

 □

2.2 Solution for the fractional diffusion equation with conformable derivative

Assume that Problem (1) satisfying (2) and (3) (i.e. the direct problem) has a solution $\mathfrak {u}$ as follows:

$$\begin{aligned} \mathfrak {u}(\textrm {t}, \textrm {x}) = \sum_{\textrm {m}=1}^{\infty } \bigl\langle \mathfrak {u}(\textrm {t}, \textrm {x}), \textrm{e}_{\textrm {m}}(\textrm {x}) \bigr\rangle \textrm{e}_{\textrm {m}}( \textrm {x}). \end{aligned}$$

Note

$$ \textstyle\begin{cases} \textrm {D}_{\upgamma }^{0}\langle \mathfrak {u}(\textrm {t},\textrm {x}), \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle - \widetilde{\textrm{a}}_{\textrm {m}} \langle \mathfrak {u}(\textrm {t},\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle = \langle \textrm {F}(\textrm {t},\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle , & (\textrm {t},\textrm {x}) \in (0,\mathscr{T} _{o}) \times \boldsymbol {\varOmega }, \\ \langle \mathfrak {u}(0,\textrm {x}), \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle = \langle \mathfrak {u}_{0}(\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle , & \textrm {x}\in {\boldsymbol {\varOmega }}, \end{cases} $$

(10)

where $\langle \mathfrak {B}\mathfrak {u}(\textrm {t},\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle = - \widetilde{\textrm{a}}_{\textrm {m}} \langle \mathfrak {u}(\textrm {t},\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \rangle$. Using the result in [14] and [22], the solution of the latter problem is

$$\begin{aligned} \mathfrak {u}(\textrm {t}, \textrm {x}) ={}& \sum_{\textrm {m}=1}^{\infty } \biggl[ \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}^{\upgamma }}{\upgamma } \biggr) \bigl\langle \mathfrak {u}_{0}(\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \bigr\rangle \\ &{} + \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}} _{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \bigl\langle \textrm {F}(\tau ,\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \bigr\rangle \,d\tau \biggr] \textrm{e}_{\textrm {m}}(\textrm {x}). \end{aligned}$$

Let $\textrm {t}= \mathscr{T}_{o}$ and we get (with $\textrm {h}_{\textrm {m}}= \langle \textrm {h}, \textrm{e}_{\textrm {m}}\rangle$ and $\textrm {F}_{\textrm {m}}(\tau )=\langle \textrm {F}(\tau ,. ) , \textrm{e}_{\textrm {m}}\rangle$)

$$\begin{aligned} \textrm {h}_{\textrm {m}} ( \textrm {x}) ={}& \exp \biggl(- \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \bigl\langle \mathfrak {u}_{0}(\textrm {x}) , \textrm{e}_{\textrm {m}}(\textrm {x}) \bigr\rangle \\ &{} + \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \textrm {F}_{\textrm {m}} (\tau )\,d\tau . \end{aligned}$$

This implies that

$$\begin{aligned} \mathfrak {u}(\textrm {t}, \textrm {x}) ={}& \sum_{\textrm {m}=1}^{\infty } \biggl[ \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \\ &{} \times \biggl( \textrm {h}_{\textrm {m}} - \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \textrm {F}_{\textrm {m}} (\tau )\,d\tau \biggr) \\ &{} + \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \textrm {F}_{\textrm {m}} (\tau )\,d\tau \biggr] \textrm{e}_{\textrm {m}}(\textrm {x}). \end{aligned}$$

(11)

Let

$$\begin{aligned} &\mathcal {A}_{\upgamma }^{1} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}:= \sum_{\textrm {m}=1}^{\infty } \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \langle \textrm {v}, \textrm{e} _{\textrm {m}} \rangle \textrm{e}_{\textrm {m}}, \\ &\mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {v}:= \sum _{\textrm {m}=1}^{\infty } \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}}, \end{aligned}$$

for $\textrm {v}\in L^{2}(\boldsymbol {\varOmega })$, and $0 \le \tau \le \textrm {t}\le \mathscr{T}_{o}$. Then it follows from (11) that

$$\begin{aligned} \mathfrak {u}(\textrm {t}) = \mathcal {A}_{\upgamma }^{1} (\textrm {t}, \mathscr{T}_{o}) \bigl[ \textrm {h}- \mathcal {A}_{\upgamma } ^{2} (\mathscr{T}_{o}) \textrm {F}(\textrm {t}) \bigr] + \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {F}(\textrm {t}) . \end{aligned}$$

Recall for any $\textrm {n}> 0$, there exists a positive constant $\mathcal {P}_{1,\textrm {n}}$ (from elementary calculus note we can take $\mathcal {P}_{1,\textrm {n}}$ to be $\textrm {n}^{\textrm {n}} \exp (-\textrm {n})$) such that

$$\begin{aligned} \exp (-z ) \le \mathcal {P}_{1,\textrm {n}} z^{-\textrm {n}},\quad z \ge 0 \end{aligned}$$

(12)

and if $0 < \textrm {s}< 1$, there exists a positive constant $\mathcal {P}_{2,\textrm {s}}$ (we can take $\mathcal {P}_{2,s}$ to be $(1-s)^{1-s} \exp (s-1)$) such that

$$\begin{aligned} \exp (-z) \le \mathcal {P}_{2,\textrm {s}} z^{\textrm {s}-1 },\quad z \ge 0. \end{aligned}$$

(13)

Lemma 2.3

Given $0 \le \tau \le \mathscr{T}_{o}$and $\boldsymbol {\varOmega }\subset \mathbb{R}^{d}$for any $1 \le d \le 3$.

• (a) If $\textrm {w}\in L^{2}(\boldsymbol {\varOmega })$, then $\| \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\| \in L^{2}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \Vert \textrm {w}\Vert _{L^{2}(\boldsymbol {\varOmega })} \quad \textit{where } \mathcal{R}= \Biggl( \sum_{\textrm {m}=1}^{\infty } \frac{1}{C^{2} \textrm {m}^{ \frac{4}{d}}} \Biggr)^{\frac{1}{2}}. \end{aligned}$$

• (b) If $\textrm {w}\in L^{2}(\boldsymbol {\varOmega })$for $0 < \textrm {n}\neq 1$, then $\| \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\| \in \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })}^{2} \le \mathcal {P}_{1,\textrm {n}} \frac{\mathscr{T}_{o}^{\upgamma (1- \textrm {n})}}{\upgamma (1- \textrm {n})} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {w}(\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau . \end{aligned}$$

• (c) If $\textrm {w}\in \mathcal {H}^{\textrm {q}+1}(\boldsymbol {\varOmega })$for $0 < \textrm {q}< 1$and $0 \le \textrm {t}_{1} < \textrm {t}_{2} \le \mathscr{T}_{o}$, then $\| \mathcal {A}_{\upgamma } ^{2} (\textrm {t}) \textrm {w}\| \in L^{2}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} & \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{2} (\textrm {t}_{1} ) - \mathcal {A}_{\upgamma }^{2} (\textrm {t}_{2} ) \bigr) \textrm {w}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ & \quad \le \biggl( \mathcal {P}_{2,\textrm {q}} \frac{ \vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2}^{\upgamma } \vert ^{2}}{\upgamma ^{\textrm {q}+1}} \frac{\mathscr{T}_{o}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma } \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega })} ^{2}\,d\tau \biggr)^{\frac{1}{2}} \\ &\qquad {} + \biggl( \frac{\textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau \biggr)^{\frac{1}{2}} . \end{aligned}$$

Proof

(a) Note

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} &= \sqrt{ \sum_{\textrm {m}=1}^{\infty } \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \bigl\langle \textrm {w}(\tau ), \textrm{e}_{\textrm {m}} \bigr\rangle \,d \tau \biggr]^{2}} \\ &\le \Vert \textrm {w}\Vert _{L^{2}(\boldsymbol {\varOmega })} \sqrt{ \sum _{\textrm {m}=1}^{\infty } \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr]^{2}}. \end{aligned}$$

(14)

By the change variable $\upxi = \frac{\tau ^{\upgamma }}{\upgamma }$, we obtain

$$\begin{aligned} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau &= \int _{0}^{\frac{\textrm {t}^{\upgamma }}{\upgamma }} \exp \biggl( - \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } }{\upgamma } + \widetilde{ \textrm{a}}_{\textrm {m}}\upxi \biggr)\,d\upxi \\ &= \frac{1 - \exp ( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{\textrm {t}^{\upgamma }}{\upgamma } )}{\widetilde{\textrm{a}}_{\textrm {m}}} \le \frac{1}{ \widetilde{\textrm{a}}_{\textrm {m}}}. \end{aligned}$$

(15)

Since $\widetilde{\textrm{a}}_{\textrm {m}}\ge C \textrm {m}^{\frac{2}{d}}$, where d is the dimension of the domain Ω, we know that

$$\begin{aligned} \sum_{\textrm {m}=1}^{\infty } \frac{1}{\widetilde{\textrm{a}}_{\textrm {m}}^{2} } \le \sum_{\textrm {m}=1}^{\infty } \frac{1}{C^{2} \textrm {m}^{\frac{4}{d}}} = \mathcal{R}^{2}. \end{aligned}$$

(16)

Combining (14), (15) and (16), we deduce that

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \Vert \textrm {w}\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

(b) For $\textrm {n}> 0$, using (12) we have

$$\begin{aligned} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \le \mathcal {P}_{1,\textrm {n}} \widetilde{\textrm{a}}_{\textrm {m}} ^{-\textrm {n}} \upgamma ^{\textrm {n}} \bigl( \textrm {t}^{\upgamma } - \tau ^{\upgamma } \bigr)^{-\textrm {n}}. \end{aligned}$$

Therefore

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {w}\bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })}^{2} ={}& \sum_{\textrm {m}=1}^{\infty } \widetilde{\textrm{a}}_{\textrm {m}}^{\textrm {n}} \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \bigl\langle \textrm {w}(\tau ), \textrm{e}_{\textrm {m}} \bigr\rangle \,d\tau \biggr]^{2} \\ \le{}& \sum_{\textrm {m}=1}^{\infty }\widetilde{ \textrm{a}}_{\textrm {m}}^{\textrm {n}} \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr] \\ &{} \times \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \bigl\langle \textrm {w}(\tau ), \textrm{e}_{\textrm {m}} \bigr\rangle ^{2}\,d\tau \biggr] \\ \le{}& \mathcal {P}_{1,\textrm {n}} \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl( \textrm {t}^{\upgamma } - \tau ^{\upgamma } \bigr)^{-\textrm {n}}\,d\tau \biggr] \sum_{\textrm {m}=1}^{\infty } \biggl[ \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\langle \textrm {w}(\tau ), \textrm{e}_{\textrm {m}} \bigr\rangle ^{2}\,d\tau \biggr] \\ \le{}& \mathcal {P}_{1,\textrm {n}} \frac{\mathscr{T}_{o}^{\upgamma (1- \textrm {n})}}{\upgamma (1- \textrm {n})} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {w}(\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau , \end{aligned}$$

where we have noted that

$$\begin{aligned} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl( \textrm {t}^{\upgamma } - \tau ^{\upgamma } \bigr)^{-\textrm {n}}\,d\tau &= \frac{1}{\upgamma } \int _{0}^{\textrm {t}} \bigl( \textrm {t}^{\upgamma } - \upxi \bigr)^{-\textrm {n}}\,d\upxi \\ &= \frac{\textrm {t}^{\upgamma (1- \textrm {n})}}{\upgamma (1- \textrm {n})} \le \frac{\mathscr{T}_{o}^{\upgamma (1- \textrm {n})}}{\upgamma (1- \textrm {n})}. \end{aligned}$$

(c) Since $0 \le \textrm {t}_{1} < \textrm {t}_{2} \le \mathscr{T}_{o}$ we have

$$\begin{aligned} & \bigl( \mathcal {A}_{\upgamma }^{2} (\textrm {t}_{1} ) - \mathcal {A}_{\upgamma }^{2} (\textrm {t}_{2} ) \bigr) \textrm {w}\\ &\quad = \sum_{\textrm {m}=1}^{\infty } \int _{0}^{\textrm {t}_{1}} \biggl[ \exp \biggl( - \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr) - \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}_{2}^{\upgamma }}{\upgamma } \biggr) \biggr] \tau ^{\upgamma - 1} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \tau ^{\upgamma }}{\upgamma } \biggr) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}} \\ &\qquad {} - \sum_{\textrm {m}=1}^{\infty } \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } \biggr) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}} \\ &\quad = \mathfrak{D}_{1} (\textrm {m},\textrm {t},\upgamma ) - \mathfrak{D}_{2} ( \textrm {m},\textrm {t},\upgamma ), \end{aligned}$$

where $\mathfrak{D}_{2} (\textrm {m},\textrm {t},\upgamma )= \sum_{\textrm {m}=1}^{\infty } \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp ( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } ) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}}$. Also note $| \exp (-a ) - \exp (-b) | \le |a - b| \max \{ \exp (-a ), \exp (-b ) \}$ and for $\textrm {t}_{1} < \textrm {t}_{2}$ then $\textrm {t}_{1}^{\upgamma } < \textrm {t}_{2}^{\upgamma }$ and

$$\begin{aligned} \biggl\vert \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr) - \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}_{2}^{\upgamma }}{\upgamma } \biggr) \biggr\vert \le \frac{\widetilde{\textrm{a}}_{\textrm {m}}}{\upgamma } \bigl\vert \textrm {t}_{1} ^{\upgamma } - \textrm {t}_{2}^{\upgamma } \bigr\vert \exp \biggl( -\widetilde{\textrm{a}} _{\textrm {m}} \frac{ \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr), \end{aligned}$$

and this together with Hölder’s inequality yields

$$\begin{aligned} & \bigl\Vert \mathfrak{D}_{1} (\textrm {m},\textrm {t},\upgamma ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2} \\ &\quad \le \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \int _{0}^{\textrm {t}_{1}} \frac{ \widetilde{\textrm{a}}_{\textrm {m}}}{\upgamma } \bigl\vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2} ^{\upgamma } \bigr\vert \tau ^{\upgamma - 1} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \tau ^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{2} \\ &\quad \leq \frac{ \vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2}^{\upgamma } \vert ^{2}}{\upgamma ^{2}} \sum_{\textrm {m}=1}^{\infty } \widetilde{\textrm{a}}_{\textrm {m}}^{2} \biggl[ \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \exp \biggl( \widetilde{\textrm{a}} _{\textrm {m}}\frac{ \tau ^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr] \\ &\qquad {} \times \biggl[ \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \tau ^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}\,d\tau \biggr]. \end{aligned}$$

For $0 < \textrm {q}< 1$, using (13) we obtain

$$\begin{aligned} \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \tau ^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \biggr) \le \mathcal {P}_{2,\textrm {q}} \widetilde{\textrm{a}}_{\textrm {m}} ^{\textrm {q}-1 } \upgamma ^{1-\textrm {q}} \bigl( \textrm {t}_{1}^{\upgamma } - \tau ^{\upgamma } \bigr)^{\textrm {q}- 1}. \end{aligned}$$

This implies that

$$\begin{aligned} & \bigl\Vert \mathfrak{D}_{1} (\textrm {m},\textrm {t},\upgamma ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2} \\ &\quad \le \mathcal {P}_{2,\textrm {q}} \frac{ \vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2}^{\upgamma } \vert ^{2}}{\upgamma ^{\textrm {q}+1}} \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \bigl( \textrm {t}_{1}^{\upgamma } - \tau ^{\upgamma } \bigr)^{\textrm {q}- 1}\,d\tau \sum_{\textrm {m}=1}^{\infty }\widetilde{ \textrm{a}}_{\textrm {m}}^{\textrm {q}+1 } \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \bigl\vert \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}\,d\tau . \end{aligned}$$

We note that

$$\begin{aligned} \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \bigl( \textrm {t}_{1}^{\upgamma } - \tau ^{\upgamma } \bigr)^{\textrm {q}- 1}\,d\tau = \frac{1}{\upgamma } \int _{0}^{\textrm {t}_{1}^{\upgamma }} \bigl( \textrm {t}^{\upgamma } - \upxi \bigr)^{\textrm {q}- 1}\,d\upxi = \frac{\textrm {t}_{1}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma } \le \frac{\mathscr{T}_{o}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma }. \end{aligned}$$

Therefore

$$\begin{aligned} \bigl\Vert \mathfrak{D}_{1} (\textrm {m},\textrm {t},\upgamma ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2} \le \mathcal {P}_{2,\textrm {q}} \frac{ \vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2}^{\upgamma } \vert ^{2}}{\upgamma ^{\textrm {q}+1}} \frac{\mathscr{T}_{o}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma } \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega })}^{2}\,d\tau . \end{aligned}$$

Using Hölder’s inequality, we have

$$\begin{aligned} & \bigl\Vert \mathfrak{D}_{2} (\textrm {m},\textrm {t},\upgamma ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } \biggr) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2} \\ &\quad = \sum_{\textrm {m}=1}^{\infty } \biggl[ \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } \biggr) \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \biggr]^{2} \\ &\quad \le \sum_{\textrm {m}=1}^{\infty } \biggl[ \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr] \biggl[ \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}_{2}^{\upgamma }- \tau ^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}\,d\tau \biggr] \\ &\quad \le \sum_{\textrm {m}=1}^{\infty } \biggl[ \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1}\,d\tau \biggr] \biggl[ \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \bigl\vert \langle \textrm {w}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}\,d\tau \biggr] \\ &\quad \le \frac{\textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \int _{\textrm {t}_{1}} ^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau . \end{aligned}$$

From the above results, we get

$$\begin{aligned} & \bigl\Vert \bigl(\mathcal {A}_{\upgamma }^{2} (\textrm {t}_{1} ) - \mathcal {A}_{\upgamma }^{2} (\textrm {t}_{2} ) \bigr) \textrm {w}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ & \quad \le \biggl( \mathcal {P}_{2,\textrm {q}} \frac{ \vert \textrm {t}_{1}^{\upgamma } - \textrm {t}_{2}^{\upgamma } \vert ^{2}}{\upgamma ^{\textrm {q}+1}} \frac{\mathscr{T}_{o}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma } \int _{0}^{\textrm {t}_{1}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega })} ^{2}\,d\tau \biggr)^{\frac{1}{2}} \\ &\qquad {} + \biggl( \frac{\textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma }}{\upgamma } \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \tau ^{\upgamma - 1} \Vert \textrm {w}\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau \biggr)^{\frac{1}{2}} . \end{aligned}$$

 □

2.3 Ill-posedness of determining initial data and stability estimate

Recall that the solution $\mathfrak {u}(\textrm {t},\textrm {x})$ of Problem (4) (i.e. the inverse problem) is given by (11). Let

$$\begin{aligned} \mathfrak {g}(\textrm {x}) := \mathfrak {u}(0,\textrm {x}) = \sum_{\textrm {m}=1}^{\infty } \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \boldsymbol {\varUpsilon }_{\textrm {m}} \textrm{e}_{\textrm {m}}(\textrm {x}), \end{aligned}$$

(17)

where

$$\begin{aligned} \boldsymbol {\varUpsilon }_{\textrm {m}} = \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle ,\quad \text{with } \boldsymbol {\varUpsilon }= \textrm {h}- \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}(\tau ) . \end{aligned}$$

(18)

In this paper, our main purpose is to determine the initial value $\mathfrak {g}(\textrm {x})$ from the final data $\textrm {h}(\textrm {x})$ and the source term $\textrm {F}(\textrm {t},\textrm {x})$. We can find $\mathfrak {g}(\textrm {x})$ by solving an operator equation as follows:

$$\begin{aligned} \mathcal {K}\mathfrak {g}= \boldsymbol {\varUpsilon }, \end{aligned}$$

where $\mathcal {K}: L^{2}(\boldsymbol {\varOmega })\to L^{2}(\boldsymbol {\varOmega })$ is the integral operator defined by

$$\begin{aligned} (\mathcal {K}\mathfrak {g}) (\textrm {x}) := \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}}= \int _{\boldsymbol {\varOmega }} \varrho ( \upxi , \textrm {x}) \mathfrak {g}(\upxi )\,d\xi , \end{aligned}$$

with kernel $\varrho (\cdot , \cdot )$ given by

$$\begin{aligned} \varrho (\upxi , \textrm {x}) := \sum_{\textrm {m}=1}^{\infty } \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \textrm{e}_{\textrm {m}}(\upxi ) \textrm{e}_{\textrm {m}}(\textrm {x}). \end{aligned}$$

Since $\varrho (\upxi , \textrm {x}) = \varrho (\textrm {x}, \upxi )$, we know that the operator $\mathcal {K}$ is self-adjoint. Assume that $\boldsymbol {\varUpsilon }\in L^{2}(\boldsymbol {\varOmega })$.

Lemma 2.4

Let $\textrm {h}\in L^{2}(\boldsymbol {\varOmega })$and $\textrm {F}\in L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$. Thenϒas in (18) belongs to $L^{2}(\boldsymbol {\varOmega })$and

$$\begin{aligned} \Vert \boldsymbol {\varUpsilon }\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \Vert \textrm {h}\Vert _{L^{2}(\boldsymbol {\varOmega })} + \mathcal{R} \Vert \textrm {F}\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

Proof

From the definition of ϒ, we get

$$\begin{aligned} \Vert \boldsymbol {\varUpsilon }\Vert _{L^{2}(\boldsymbol {\varOmega })} &\le \bigl\Vert \textrm {h}- \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}(\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\le \Vert \textrm {h}\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}(\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} . \end{aligned}$$

Using Lemma 2.3, we deduce that

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}(\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \bigl\Vert \textrm {F}(\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \Vert \textrm {F}\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

(19)

Therefore

$$\begin{aligned} \Vert \boldsymbol {\varUpsilon }\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \Vert \textrm {h}\Vert _{L^{2}(\boldsymbol {\varOmega })} + \mathcal{R} \Vert \textrm {F}\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

This completes the proof. □

Therefore, $\mathcal {K}: L^{2}(\boldsymbol {\varOmega })\to L^{2}(\boldsymbol {\varOmega })$ is compact operator of infinite rank. Hence $\mathcal {K}$ does not have a continuous inverse [24].

To illustrate the ill-posedness of the backward problem, we give an example. Let $(\textrm {h}, \textrm {F})= (0,0)$ and $(\overline{\textrm {h}}, \overline{\textrm {F}}) = ( \frac{1}{\sqrt{\widetilde{\textrm{a}}_{\textrm{l}}}} \textrm{e}_{\textrm{l}},\frac{1}{\sqrt{\widetilde{\textrm{a}}_{ \textrm{l}}}}\textrm{e}_{\textrm{l}} )$. It is easy to see that

$$\begin{aligned} \Vert \overline{\textrm {h}} - \textrm {h}\Vert = \frac{1}{\sqrt{\widetilde{\textrm{a}} _{\textrm{l}}}},\quad \text{and}\quad \Vert \overline{\textrm {F}} - \textrm {F}\Vert = \frac{1}{\sqrt{ \widetilde{\textrm{a}}_{\textrm{l}}}}. \end{aligned}$$

Hence

$$\begin{aligned} \lim_{\textrm{l} \rightarrow \infty } \Vert \overline{\textrm {h}} - \textrm {h}\Vert = 0 , \quad \text{and} \quad \lim_{\textrm{l} \rightarrow \infty } \Vert \overline{\textrm {F}} - \textrm {F}\Vert =0, \end{aligned}$$

(20)

so $(\overline{\textrm {h}}, \overline{\textrm {F}})$ is an approximation of $(\textrm {h}, \textrm {F})$ when l is large enough. Using $( \overline{\textrm {h}}, \overline{\textrm {F}})$, we get the corresponding initial data $\overline{\mathfrak {g}}$ and the equation $\overline{\boldsymbol {\varUpsilon }}$ as follows:

$$\begin{aligned} &\overline{\boldsymbol {\varUpsilon }}(\textrm {x})= \sum_{\textrm {m}=1}^{\infty } \biggl[ \overline{\textrm {h}} _{\textrm {m}} - \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \overline{\textrm {F}}_{\textrm {m}} (\tau )\,d\tau \biggr] \textrm{e}_{\textrm {m}}( \textrm {x}), \\ &\overline{\mathfrak {g}}(\textrm {x})= \sum_{\textrm {m}=1}^{\infty } \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ \overline{\textrm {h}}_{\textrm {m}} - \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \overline{ \textrm {F}}_{\textrm {m}} (\tau )\,d\tau \biggr] \textrm{e}_{\textrm {m}}(\textrm {x}). \end{aligned}$$

From Parseval’s equality and (15), we get

$$\begin{aligned} \Vert \overline{\boldsymbol {\varUpsilon }} - \boldsymbol {\varUpsilon }\Vert ^{2} &= \sum _{\textrm {m}=1}^{\infty } \biggl[ \langle \overline{\textrm {h}}_{\textrm {m}} - \overline{\textrm {h}} , \textrm{e}_{\textrm {m}} \rangle - \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \langle \overline{\textrm {F}}_{\textrm {m}} - \textrm {F}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \biggr] ^{2} \\ &= \biggl[ \frac{1}{\sqrt{\widetilde{\textrm{a}}_{\textrm{l}}}} - \frac{1}{\sqrt{ \widetilde{\textrm{a}}_{\textrm{l}}}} \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm{l}} \frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr]^{2} \\ &= \frac{1}{\widetilde{\textrm{a}}_{\textrm{l}}} \biggl[ 1 - \frac{1 - \exp ( - \widetilde{\textrm{a}}_{\textrm{l}} \frac{\mathscr{T} _{o}^{\upgamma }}{\upgamma } )}{\widetilde{\textrm{a}}_{\textrm{l}}} \biggr]^{2}. \end{aligned}$$

This gives

$$\begin{aligned} \lim_{\textrm{l} \rightarrow \infty } \Vert \overline{\boldsymbol {\varUpsilon }} - \boldsymbol {\varUpsilon }\Vert &= 0 . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} & \Vert \overline{\mathfrak {g}} - \mathfrak {g}\Vert ^{2} \\ &\quad = \sum_{\textrm {m}=1}^{\infty }\exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ \langle \overline{\textrm {h}}_{\textrm {m}} - \overline{\textrm {h}} , \textrm{e}_{\textrm {m}} \rangle \\ &\qquad {} - \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \langle \overline{\textrm {F}}_{\textrm {m}} - \textrm {F}, \textrm{e}_{\textrm {m}} \rangle \,d\tau \biggr] ^{2} \\ &\quad = \exp \biggl(\widetilde{\textrm{a}}_{\textrm{l}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ \frac{1}{\sqrt{ \widetilde{\textrm{a}}_{\textrm{l}}}} - \frac{1}{\sqrt{ \widetilde{\textrm{a}}_{\textrm{l}}}} \int _{0}^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \exp \biggl( -\widetilde{\textrm{a}}_{\textrm{l}} \frac{ \mathscr{T}_{o}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr)\,d\tau \biggr]^{2} \\ &\quad = \exp \biggl( \widetilde{\textrm{a}}_{\textrm{l}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \frac{1}{\widetilde{\textrm{a}} _{\textrm{l}}} \biggl[ 1 - \frac{1 - \exp ( - \widetilde{\textrm{a}}_{\textrm{l}} \frac{\mathscr{T}_{o}^{\upgamma }}{\upgamma } )}{\widetilde{\textrm{a}}_{ \textrm{l}}} \biggr]^{2}, \end{aligned}$$

so

$$\begin{aligned} \lim_{\textrm{l} \rightarrow +\infty } \Vert \overline{\mathfrak {g}} - \mathfrak {g}\Vert &= + \infty . \end{aligned}$$

(21)

We conclude that the backward problem is ill-posed in the Hadamard sense. Hence a regularization method is necessary. We will use the Landweber method to deal with the ill-posed problem. Before doing that, we impose an a priori bound on the initial data; that is,

$$\begin{aligned} \sum_{\textrm {m}=1}^{\infty }\exp \biggl( 2\textrm {r}\widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} \le \textbf{E}^{2}, \quad \text{for any } \textrm {r}\ge 0, \end{aligned}$$

(22)

where E is a positive constant. The a priori bound of the exact solution is necessary for any ill-posed problem, otherwise, the rate of convergence is very slow or the regularization solution is not convergent (see [12]).

3 Landweber regularization method and regularity of the regularized solution

In this section, we present a regularized problem by using the Landweber regularization method, and also we consider the well- posedness of the regularized solution. From [17], the operator equation $\mathcal {K}\mathfrak {g}= \boldsymbol {\varUpsilon }$ is equivalent to the following equation:

$$\begin{aligned} \mathfrak {g}= \bigl( I - \upmu \mathcal {K}^{*} \mathcal {K}\bigr) \mathfrak {g}+ \upmu \mathcal {K}^{*} \boldsymbol {\varUpsilon }, \end{aligned}$$

(23)

for any $\upmu > 0$. Here, $\mathcal {K}^{*}$ is the adjoint operator of $\mathcal {K}$, and $\upmu >0$ satisfies $0 < \upmu < \frac{1}{ \Vert \mathcal {K}\Vert ^{2}}$. The iterative implementation of the Landweber method was constructed in [18]. Denote the Landweber regularization solution by

$$\begin{aligned} \mathfrak {g}_{\textrm {k}, \upalpha }(\textrm {x})= \sum_{\textrm {m}=1}^{\infty } \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}}, \end{aligned}$$

(24)

and the Landweber regularization solution with noisy data by

$$\begin{aligned} \mathfrak {g}_{\textrm {k}, \upalpha }^{\upepsilon }(\textrm {x})= \sum_{\textrm {m}=1}^{\infty } \exp \biggl( \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \bigl\langle \boldsymbol {\varUpsilon }^{\upepsilon }, \textrm{e}_{\textrm {m}} \bigr\rangle \textrm{e}_{\textrm {m}}, \end{aligned}$$

(25)

where $\upalpha \in ( \frac{1}{2},1]$ is called the fractional parameter, and $\textrm {k}= 1,2,3,\ldots$ is a regularization parameter. When $\upalpha = 1$, this is the classical Landweber method.

Hence, we get the Landweber regularization solution of Problem (4) (i.e. the inverse problem):

$$\begin{aligned} \mathfrak {u}_{\textrm {k}, \upalpha } (\textrm {t}, \textrm {x}) ={}& \sum_{\textrm {m}=1}^{\infty } \biggl[ \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \\ &{} \times \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \boldsymbol {\varUpsilon }, \textrm{e} _{\textrm {m}} \rangle \\ &{} + \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \tau ^{\upgamma }}{\upgamma } \biggr) \textrm {F}_{\textrm {m}} (\tau )\,d\tau \biggr] \textrm{e}_{\textrm {m}}(\textrm {x}). \end{aligned}$$

(26)

Let us consider the operator

$$\begin{aligned} \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}:={}& \sum_{\textrm {m}=1}^{\infty } \exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \\ &{} \times \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}}, \end{aligned}$$

for $\textrm {v}\in L^{2}(\boldsymbol {\varOmega })$, and $0 \le \textrm {t}\le \mathscr{T}_{o}$.

Lemma 3.1

Given $\upmu > 0$and $\textrm {k}\in \mathbb{N}$with $\textrm {k}\ge 1$.

• (a) If $\textrm {v}\in L^{2}(\boldsymbol {\varOmega })$, then $\| \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\| \in L^{2}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \Vert \textrm {v}\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

• (b) If $\textrm {v}\in L^{2}(\boldsymbol {\varOmega })$for $\textrm {n}>0$, then $\| \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\| \in \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{ \frac{1}{2}} \mathcal {P}_{1, \textrm {n}}^{\frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \textrm {t}^{-\frac{\textrm {n}\upgamma }{2}} \Vert \textrm {v}\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

• (c) If $\textrm {v}\in \mathcal {H}^{\textrm {s}}(\boldsymbol {\varOmega })$for $0 < \textrm {s}< 1$and $0 \le \textrm {t}_{1} < \textrm {t}_{2} \le \mathscr{T}_{o}$, then $\| \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\| \in L^{2}(\boldsymbol {\varOmega })$and

  $$\begin{aligned} \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}_{1} , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}_{2}, \mathscr{T}_{o}) \bigr) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,\textrm {s}}}{\textrm {s}\upgamma ^{\textrm {s}}} \bigl[ \textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma } \bigr]^{\textrm {s}} \Vert \textrm {v}\Vert _{\mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })}. \end{aligned}$$

Proof

(a) First, using Lemma 2.2, we obtain

$$\begin{aligned} & \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \Biggl\Vert \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \sum _{\textrm {m}=1} ^{\infty }\exp \biggl( - \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } }{\upgamma } \biggr) \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \Vert \textrm {v}\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

(b) For $\textrm {n}> 0$, using (12) we have

$$\begin{aligned} & \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {v}\bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{\mathcal {H}^{\textrm {n}}} \\ &\quad \le \Biggl\Vert \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \sum _{\textrm {m}=1} ^{\infty }\exp \biggl( - \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } }{\upgamma } \biggr) \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \sqrt{ \sum _{\textrm {m}=1} ^{\infty }\widetilde{\textrm{a}}_{\textrm {m}}^{\textrm {n}} \exp \biggl( - 2 \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } }{\upgamma } \biggr) \bigl\vert \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} } \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1, \textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \textrm {t}^{- \frac{\textrm {n}\upgamma }{2}} \Vert \textrm {v}\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

(c) From the definition of $\mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T} _{o})$, we get

$$\begin{aligned} & \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}_{1} , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}_{2}, \mathscr{T}_{o}) \bigr) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl[ \exp \biggl( - \frac{ \widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{1}^{\upgamma } }{\upgamma } \biggr) - \exp \biggl( - \frac{ \widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{2}^{\upgamma } }{\upgamma } \biggr) \biggr] \exp \biggl( \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \\ &\qquad {} \times \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Using Lemma 2.2, we get

$$\begin{aligned} & \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}_{1} , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}_{2}, \mathscr{T}_{o}) \bigr) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \Biggl( \upmu \textrm {k}\sum_{\textrm {m}=1}^{\infty } \biggl[ \exp \biggl( \frac{ -\widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{1}^{\upgamma } }{\upgamma } \biggr) - \exp \biggl( \frac{-\widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{2}^{\upgamma } }{\upgamma } \biggr) \biggr] ^{2} \bigl\vert \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. \end{aligned}$$

For $0 < \textrm {s}< 1$, using (13) we obtain

$$\begin{aligned} \exp \biggl( \frac{ -\widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{1}^{\upgamma } }{\upgamma } \biggr) - \exp \biggl( \frac{-\widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}_{2}^{\upgamma } }{\upgamma } \biggr) &= \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \exp \biggl(- \frac{ \widetilde{\textrm{a}} _{\textrm {m}}\textrm {t}^{\upgamma } }{\upgamma } \biggr)\,d\biggl( \frac{ \widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}^{\upgamma } }{\upgamma } \biggr) \\ &\le \mathcal {P}_{2,\textrm {s}} \int _{\textrm {t}_{1}}^{\textrm {t}_{2}} \biggl( \frac{ \widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}^{\upgamma } }{\upgamma } \biggr)^{\textrm {s}-1}\,d\biggl( \frac{ \widetilde{\textrm{a}}_{\textrm {m}}\textrm {t}^{\upgamma } }{\upgamma } \biggr) \\ &\le \frac{\mathcal {P}_{2,\textrm {s}}}{\textrm {s}} \biggl( \frac{ \widetilde{\textrm{a}} _{\textrm {m}}}{\upgamma } \biggr)^{\textrm {s}} \bigl[ \textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma } \bigr]^{\textrm {s}}. \end{aligned}$$

From the above results, we deduce that

$$\begin{aligned} & \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}_{1} , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}_{2}, \mathscr{T}_{o}) \bigr) \textrm {v}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,\textrm {s}}}{\textrm {s}\upgamma ^{\textrm {s}}} \bigl[ \textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma } \bigr]^{\textrm {s}} \Biggl( \sum_{\textrm {m}=1}^{\infty } \widetilde{\textrm{a}}_{\textrm {m}}^{\textrm {s}} \bigl\vert \langle \textrm {v}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} \Biggr)^{ \frac{1}{2}} \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,\textrm {s}}}{\textrm {s}\upgamma ^{\textrm {s}}} \bigl[ \textrm {t}_{2}^{\upgamma } - \textrm {t}_{1}^{\upgamma } \bigr]^{\textrm {s}} \Vert \textrm {v}\Vert _{\mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })}. \end{aligned}$$

 □

The Landweber regularization solution of Problem (4) can be transformed into the form

$$\begin{aligned} \mathfrak {u}_{\textrm {k}, \upalpha } (\textrm {t}) = \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \bigl[ \textrm {h}- \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}(\textrm {t}) \bigr] + \mathcal {A}_{\upgamma } ^{2} (\textrm {t}) \textrm {F}(\textrm {t}) . \end{aligned}$$

(27)

and the Landweber regularization solution of Problem (4) with noisy data:

$$\begin{aligned} \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } (\textrm {t}) = {}& \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T} _{o}) \bigl[ \textrm {h}^{\upepsilon } - \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr] + \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {F}^{\upepsilon } (\textrm {t}) \\ = {}& \varXi _{1} (\textrm {t}) + \varXi _{2} (\textrm {t}) + \varXi _{3} (\textrm {t}). \end{aligned}$$

(28)

Next, we consider the regularity of the solution $\mathfrak {u}_{\textrm {k}, \upalpha } ^{\upepsilon }$. We give a result which establishes regularity of the regularized solution.

Theorem 3.1

• (a) Let $\textrm {h}^{\upepsilon } \in L^{2}(\boldsymbol {\varOmega })$and $\textrm {F}\in L^{\infty }(0,\mathscr {T}_{o};L ^{2}(\boldsymbol {\varOmega }))$. If $\boldsymbol {\varOmega }\subset \mathbb{R}^{d}$for any $1 \le d \le 3$then

  $$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl( \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} + 1 \bigr) \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))}. \end{aligned}$$

• (b) Given $0 < \textrm {n}\neq 1$and $1 < \textrm {p}< \min ( \frac{2}{\textrm {n}\upgamma }, \frac{1}{1 - \upgamma } )$. Let $\textrm {h}^{\upepsilon } \in L^{2}(\boldsymbol {\varOmega })$and $\textrm {F}\in L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T} _{o};L^{2}(\boldsymbol {\varOmega })) \cap L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$. Then there exists $\mathfrak{M}$depends only $\mathscr{T}_{o}$, p, n, μ, k and (the regularized solution) $\mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \in L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })) \cap L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$such that

  $$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega }))} \le \mathfrak{M} \bigl( \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} + \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))} \bigr), \end{aligned}$$

• (c) Let $\textrm {h}^{\upepsilon } \in \mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })$for $0 < \textrm {s}< 1$and $\textrm {F}\in L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o}; \mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega }))$for $1 < \textrm {p}< \min ( \frac{2}{ \upgamma }, \frac{1}{1 - \upgamma } )$and $0 < \textrm {q}<1$. Then $\mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \in C([0,\mathscr{T}_{o}]; L^{2}(\boldsymbol {\varOmega }))$.

Proof

(a) Since $\textrm {h}^{\upepsilon } \in L^{2}(\boldsymbol {\varOmega })$ so part (a) of Lemma 3.1 yields

$$\begin{aligned} \bigl\Vert \varXi _{1} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} = \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{ \frac{1}{2}} \textrm {k}^{\frac{1}{2}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

By a similar method, we obtain

$$\begin{aligned} \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} = \bigl\Vert \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \bigl[ \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr] \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{ \frac{1}{2}} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Assume $\textrm {F}\in L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$. Now

$$\begin{aligned} \bigl\vert \bigl\langle \textrm {F}^{\upepsilon } (\textrm {t}), \textrm{e}_{\textrm {m}} \bigr\rangle \bigr\vert ^{2} \le \operatorname{ess} \sup _{0 \le \textrm {t}\le \mathscr{T}_{o}} \sum_{\textrm {m}=1} ^{\infty } \bigl\vert \bigl\langle \textrm {F}^{\upepsilon } (\textrm {t}), \textrm{e}_{\textrm {m}} \bigr\rangle \bigr\vert ^{2} = \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} ^{2}. \end{aligned}$$

Using Lemma 2.3, we deduce that

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L ^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

(29)

Hence

$$\begin{aligned} \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))}. \end{aligned}$$

On the other hand

$$\begin{aligned} \bigl\Vert \varXi _{3} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} = \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

Combining the above results, we get

$$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} &= \bigl\Vert \varXi _{1} (\textrm {t}) (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \varXi _{3} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl( \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} + 1 \bigr) \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))}. \end{aligned}$$

(30)

This implies that $\mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \in L^{\infty }(0,\mathscr {T}_{o};L ^{2}(\boldsymbol {\varOmega }))$.

(b) Since $\textrm {h}^{\upepsilon } \in L^{2}(\boldsymbol {\varOmega })$ so Lemma 3.1 with $0 < \textrm {n}$ yields

$$\begin{aligned} \bigl\Vert \varXi _{1} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} = \bigl\Vert \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \textrm {h}^{\upepsilon } \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{\frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \textrm {t}^{- \frac{\textrm {n}\upgamma }{2}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Since $\textrm {p}< \frac{2}{\textrm {n}\upgamma }$, we get

$$\begin{aligned} 1 - \frac{\textrm {p}\textrm {n}\upgamma }{2} = \frac{2 - \textrm {p}\textrm {n}\upgamma }{2} >0. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \varXi _{1} \Vert _{L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega }))} &= \biggl( \int _{0}^{\mathscr{T}_{o}} \bigl\Vert \varXi _{1} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })}^{\textrm {p}}\,d\textrm {t}\biggr)^{\frac{1}{\textrm {p}}} \\ & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \biggl( \int _{0}^{\mathscr{T}_{o}} \textrm {t}^{-\frac{\textrm {p}\textrm {n}\upgamma }{2}}\,d\textrm {t}\biggr)^{ \frac{1}{\textrm {p}}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \biggl( \frac{2}{2 - \textrm {p}\textrm {n}\upgamma } \biggr)^{\frac{1}{\textrm {p}}} \mathscr{T}_{o}^{\frac{2 - \textrm {p}\textrm {n}\upgamma }{2 \textrm {p}}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L ^{2}(\boldsymbol {\varOmega })} \\ &= \mathfrak{M}_{1}(\upmu ,\textrm {k},\textrm {p},\textrm {n}) \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}, \end{aligned}$$

(31)

where

$$\begin{aligned} \mathfrak{M}_{1}(\upmu ,\textrm {k},\textrm {p},\textrm {n}) = \upmu ^{\frac{1}{2}} \textrm {k}^{ \frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{\frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{- \frac{\textrm {n}}{2}} \biggl( \frac{2}{2 - \textrm {p}\textrm {n}\upgamma } \biggr)^{\frac{1}{\textrm {p}}} \mathscr{T}_{o}^{\frac{2 - \textrm {p}\textrm {n}\upgamma }{2 \textrm {p}}}. \end{aligned}$$

By a similar method, we obtain

$$\begin{aligned} \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} &= \bigl\Vert \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \bigl[ \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr] \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \\ & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \textrm {t}^{- \frac{\textrm {n}\upgamma }{2}} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

From (29) we obtain

$$\begin{aligned} \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \le \upmu ^{ \frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{\frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \textrm {t}^{-\frac{\textrm {n}\upgamma }{2}} \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

Therefore

$$\begin{aligned} \Vert \varXi _{2} \Vert _{L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega }))} &= \biggl( \int _{0}^{\mathscr{T}_{o}} \bigl\Vert \varXi _{2} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })}^{\textrm {p}}\,d\textrm {t}\biggr)^{\frac{1}{\textrm {p}}} \\ & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \mathcal{R} \biggl( \int _{0}^{\mathscr{T}_{o}} \textrm {t}^{- \frac{\textrm {p}\textrm {n}\upgamma }{2}}\,d\textrm {t}\biggr)^{\frac{1}{\textrm {p}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \\ & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{ \frac{1}{2}} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \mathcal{R} \biggl( \frac{2}{2 - \textrm {p}\textrm {n}\upgamma } \biggr)^{\frac{1}{\textrm {p}}} \mathscr{T}_{o}^{\frac{2 - \textrm {p}\textrm {n}\upgamma }{2 \textrm {p}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L ^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \\ &= \mathfrak{M}_{2}(\upmu ,\textrm {k},\textrm {p},\textrm {n}) \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L ^{2}(\boldsymbol {\varOmega }))}, \end{aligned}$$

(32)

where

$$\begin{aligned} \mathfrak{M}_{2}(\upmu ,\textrm {k},\textrm {p},\textrm {n})= \upmu ^{\frac{1}{2}} \textrm {k}^{ \frac{1}{2}} \mathcal {P}_{1,\textrm {n}}^{\frac{1}{2}} \mathcal{R} \biggl( \frac{2}{\upgamma } \biggr)^{-\frac{\textrm {n}}{2}} \biggl( \frac{2}{2 - \textrm {p}\textrm {n}\upgamma } \biggr)^{\frac{1}{\textrm {p}}} \mathscr{T}_{o}^{\frac{2 - \textrm {p}\textrm {n}\upgamma }{2 \textrm {p}}}. \end{aligned}$$

For $\textrm {n}\neq 1$ using Lemma 2.3 we have

$$\begin{aligned} \bigl\Vert \varXi _{3} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} &= \bigl\Vert \mathcal {A}_{\upgamma } ^{2} (\textrm {t}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} \\ & \le \biggl( \frac{\mathcal {P}_{1,\textrm {n}} }{\upgamma (1- \textrm {n})} \biggr)^{\frac{1}{2}} \mathscr{T}_{o}^{\frac{\upgamma (1- \textrm {n})}{2}} \biggl( \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{2}\,d\tau \biggr)^{ \frac{1}{2}}. \end{aligned}$$

Hölder’s inequality gives

$$\begin{aligned} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{2}\,d\tau &= \biggl( \int _{0}^{\textrm {t}} \tau ^{\textrm {p}(\upgamma - 1)}\,d\tau \biggr)^{ \frac{1}{\textrm {p}}} \biggl( \int _{0}^{\textrm {t}} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{\frac{2 \textrm {p}}{\textrm {p}- 1}}\,d\tau \biggr)^{\frac{\textrm {p}- 1}{\textrm {p}}} \\ &\le \frac{\mathscr{T}_{o}^{\upgamma - 1 + \frac{1}{\textrm {p}}}}{(\upgamma \textrm {p}- \textrm {p}+1 )^{\frac{1}{\textrm {p}}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0, \mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}^{2} . \end{aligned}$$

From the above results we have

$$\begin{aligned} \bigl\Vert \varXi _{3} (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega })} &\le \biggl( \frac{\mathcal {P}_{1,\textrm {n}} }{\upgamma (1- \textrm {n})} \biggr)^{\frac{1}{2}} \mathscr{T}_{o}^{\frac{\upgamma (1- \textrm {n})}{2}} \frac{\mathscr{T}_{o}^{ \frac{ (\upgamma - 1) \textrm {p}+ 1}{2\textrm {p}}}}{(\upgamma \textrm {p}- \textrm {p}+1 )^{\frac{1}{2\textrm {p}}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))} . \end{aligned}$$

Hence

$$\begin{aligned} & \Vert \varXi _{3} \Vert _{L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega }))} \\ &\quad \le \biggl( \frac{\mathcal {P}_{1,\textrm {n}} }{\upgamma (1- \textrm {n}) (\upgamma \textrm {p}- \textrm {p}+1 )^{ \frac{1}{\textrm {p}}} } \biggr)^{\frac{1}{2}} \mathscr{T}_{o}^{\upgamma - \frac{\upgamma \textrm {n}+1}{2} + \frac{3}{2 \textrm {p}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))} \\ &\quad \le \mathfrak{M}_{3}(\upmu ,\textrm {k},\textrm {p},\textrm {n}) \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}, \end{aligned}$$

(33)

where

$$\begin{aligned} \mathfrak{M}_{3}(\upmu ,\textrm {k},\textrm {p},\textrm {n}) = \biggl( \frac{\mathcal {P}_{1,\textrm {n}} }{\upgamma (1- \textrm {n}) (\upgamma \textrm {p}- \textrm {p}+1 )^{\frac{1}{\textrm {p}}} } \biggr)^{ \frac{1}{2}} \mathscr{T}_{o}^{\upgamma - \frac{\upgamma \textrm {n}+1}{2} + \frac{3}{2 \textrm {p}}}. \end{aligned}$$

Combining (31), (32) and (33) we get

$$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{\textrm {p}}(0, \mathscr{T}_{o}, \mathcal {H}^{\textrm {n}}(\boldsymbol {\varOmega }))} \le \mathfrak{M} \bigl( \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} + \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))} \bigr), \end{aligned}$$

where

$$\begin{aligned} \mathfrak{M} = \max \bigl\{ \mathfrak{M}_{i}(\upmu ,\textrm {k},\textrm {p},\textrm {n}): i = \overline{1,3} \bigr\} . \end{aligned}$$

(c) We obtain

$$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon }(\textrm {t}+ \eta ) - \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon }(\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}\le{}& \bigl\Vert \varXi _{1} (\textrm {t}+ \eta ) - \varXi _{1} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &{} + \bigl\Vert \varXi _{2} (\textrm {t}+ \eta ) - \varXi _{2} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \varXi _{3} (\textrm {t}+ \eta ) - \varXi _{3} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Now part (c) of Lemma 3.1 yields

$$\begin{aligned} \bigl\Vert \varXi _{1} (\textrm {t}+ \eta ) - \varXi _{1} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} &= \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}+ \eta , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \bigr) \textrm {h}^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,s}}{s \upgamma ^{s}} \bigl[ ( \textrm {t}+ \eta )^{\upgamma } - \textrm {t}^{\upgamma } \bigr]^{\textrm {s}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{\mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })}. \end{aligned}$$

We use the inequality $( a_{1} + a_{2} )^{\vartheta } \le a_{1} ^{ \vartheta } + a_{2} ^{\vartheta }$ for $0 < \vartheta \le 1$ to get

$$\begin{aligned} (\textrm {t}+ \eta )^{\upgamma } - \textrm {t}^{\upgamma } \le \eta ^{\upgamma } ,\quad 0< \upgamma \le 1. \end{aligned}$$

(34)

Therefore

$$\begin{aligned} \bigl\Vert \varXi _{1} (\textrm {t}+ \eta ) - \varXi _{1} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,s}}{s \upgamma ^{s}} \eta ^{\upgamma {\textrm {s}}} \bigl\Vert \textrm {h}^{\upepsilon } \bigr\Vert _{\mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })}. \end{aligned}$$

Applying a similar method gives

$$\begin{aligned} \bigl\Vert \varXi _{2} (\textrm {t}+ \eta ) - \varXi _{2} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} &= \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{3} (\textrm {t}+ \eta , \mathscr{T}_{o}) - \mathcal {A}_{\upgamma } ^{3} (\textrm {t}, \mathscr{T}_{o}) \bigr) \bigl[ \mathcal {A}_{\upgamma }^{2} (\mathscr{T} _{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr] \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,s}}{s \upgamma ^{s}} \eta ^{\upgamma {\textrm {s}}} \bigl\Vert \mathcal {A}_{\upgamma }^{2} ( \mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{\mathcal{H}^{\textrm {s}}(\boldsymbol {\varOmega })}. \end{aligned}$$

Since $0 < \textrm {s}< 1$ using Lemma 2.3 we get

$$\begin{aligned} \bigl\Vert \mathcal {A}_{\upgamma }^{2} (\mathscr{T}_{o}) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{\mathcal {H}^{\textrm {s}}(\boldsymbol {\varOmega })} &\le \biggl( \frac{\mathcal {P}_{2,\textrm {s}} }{\upgamma (1- \textrm {s})} \biggr)^{ \frac{1}{2}} \mathscr{T}_{o}^{\frac{\upgamma (1- \textrm {s})}{2}} \biggl( \int _{0} ^{\mathscr{T}_{o}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{2}\,d\tau \biggr)^{\frac{1}{2}} \\ &\le \biggl( \frac{\mathcal {P}_{2,\textrm {s}} }{\upgamma (1- \textrm {s}) (\upgamma \textrm {p}- \textrm {p}+1 )^{ \frac{1}{\textrm {p}}} } \biggr)^{\frac{1}{2}} \mathscr{T}_{o}^{\upgamma - \frac{\upgamma \textrm {s}+1}{2} + \frac{1}{2 \textrm {p}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}^{2} . \end{aligned}$$

This implies that

$$\begin{aligned} & \bigl\Vert \varXi _{2} (\textrm {t}+ \eta ) - \varXi _{2} ( \textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \frac{\mathcal {P}_{2,s}}{s \upgamma ^{s}} \eta ^{\upgamma {\textrm {s}}} \biggl( \frac{\mathcal {P}_{2,\textrm {s}} }{\upgamma (1- \textrm {s}) (\upgamma \textrm {p}- \textrm {p}+1 )^{\frac{1}{\textrm {p}}} } \biggr)^{\frac{1}{2}} \mathscr{T} _{o}^{\upgamma - \frac{\upgamma \textrm {s}+1}{2} + \frac{1}{2 \textrm {p}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}^{2} . \end{aligned}$$

Using Lemma 2.3 and $0 < \textrm {q}< 1$ so we have

$$\begin{aligned} & \bigl\Vert \varXi _{3} (\textrm {t}+ \eta ) - \varXi _{3} (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \bigl\Vert \bigl( \mathcal {A}_{\upgamma }^{2} (\textrm {t}+ \eta ) - \mathcal {A}_{\upgamma }^{2} (\textrm {t}) \bigr) \textrm {F}^{\upepsilon } (\textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \biggl( \mathcal {P}_{2,\textrm {q}} \frac{ \vert (\textrm {t}+ \eta )^{\upgamma } - \textrm {t}^{\upgamma } \vert ^{2}}{\upgamma ^{\textrm {q}+1}} \frac{\mathscr{T}_{o}^{\textrm {q}\upgamma }}{\textrm {q}\upgamma } \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{\mathcal{H} ^{\textrm {q}-2\kappa +1 }(\boldsymbol {\varOmega })}^{2}\,d\tau \biggr)^{\frac{1}{2}} \\ & \qquad {}+ \biggl( \frac{(\textrm {t}+ \eta )^{\upgamma } - \textrm {t}^{\upgamma }}{\upgamma } \int _{\textrm {t}} ^{\textrm {t}+ \eta } \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{2}\,d\tau \biggr)^{\frac{1}{2}} . \end{aligned}$$

(35)

Apply Hölder’s inequality and we have

$$\begin{aligned} \int _{0}^{\textrm {t}} \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{\mathcal{H} ^{\textrm {q}+1 }(\boldsymbol {\varOmega })}^{2}\,d\tau &\le \biggl( \int _{0}^{\textrm {t}} \tau ^{\textrm {p}(\upgamma - 1)}\,d\tau \biggr) ^{\frac{1}{\textrm {p}}} \biggl( \int _{0}^{\textrm {t}} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega })}^{\frac{2 \textrm {p}}{\textrm {p}-1}}\,d\tau \biggr)^{\frac{\textrm {p}-1}{\textrm {p}}} \\ &\le \frac{\mathscr{T}_{o}^{\upgamma -1 + \frac{1}{\textrm {p}}}}{(\upgamma \textrm {p}- \textrm {p}+1)^{ \frac{1}{\textrm {p}}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0, \mathscr{T}_{o};\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega }))}^{2} \end{aligned}$$

(36)

and

$$\begin{aligned} \int _{\textrm {t}}^{\textrm {t}+ \eta } \tau ^{\upgamma - 1} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{2}\,d\tau &\le \biggl( \int _{\textrm {t}}^{\textrm {t}+ \eta } \tau ^{\textrm {p}(\upgamma - 1)}\,d\tau \biggr)^{\frac{1}{\textrm {p}}} \biggl( \int _{\textrm {t}}^{\textrm {t}+ \eta } \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{\frac{2 \textrm {p}}{\textrm {p}-1}}\,d\tau \biggr)^{ \frac{\textrm {p}-1}{\textrm {p}}} \\ &\le \biggl( \int _{0}^{\mathscr{T}_{o}} \tau ^{\textrm {p}(\upgamma - 1)}\,d\tau \biggr)^{\frac{1}{\textrm {p}}} \biggl( \int _{0}^{\mathscr{T}_{o}} \bigl\Vert \textrm {F}^{\upepsilon } (\tau ) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}^{\frac{2 \textrm {p}}{\textrm {p}-1}}\,d\tau \biggr)^{ \frac{\textrm {p}-1}{\textrm {p}}} \\ &\le \frac{\mathscr{T}_{o}^{\upgamma -1 + \frac{1}{\textrm {p}}}}{(\upgamma \textrm {p}- \textrm {p}+1)^{ \frac{1}{\textrm {p}}}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0, \mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}^{2} . \end{aligned}$$

(37)

Combining (34), (35), (36) and (37) and we get

$$\begin{aligned} \bigl\Vert \varXi _{3} (\textrm {t}+ \eta ) - \varXi _{3} ( \textrm {t}) \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le{}& \biggl( \mathcal {P}_{2,\textrm {q}} \frac{\eta ^{2\upgamma }}{\upgamma ^{\textrm {q}+2}} \frac{ \mathscr{T}_{o}^{\textrm {q}\upgamma +\upgamma -1 + \frac{1}{\textrm {p}}}}{\textrm {q}(\upgamma \textrm {p}- \textrm {p}+1)^{ \frac{1}{\textrm {p}}}} \biggr)^{\frac{1}{2}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};\mathcal{H}^{\textrm {q}+1 }(\boldsymbol {\varOmega }))} \\ &{} + \biggl( \frac{\eta ^{\upgamma }}{\upgamma } \frac{\mathscr{T}_{o}^{\upgamma -1 + \frac{1}{\textrm {p}}}}{(\upgamma \textrm {p}- \textrm {p}+1)^{\frac{1}{\textrm {p}}}} \biggr)^{\frac{1}{2}} \bigl\Vert \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\frac{2 \textrm {p}}{\textrm {p}- 1}}(0,\mathscr{T}_{o};L^{2}(\boldsymbol {\varOmega }))}. \end{aligned}$$

Therefore, we conclude that $\mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \in C([0, \mathscr{T}_{o}]; L^{2}(\boldsymbol {\varOmega }))$. □

4 Convergence analysis and error estimate under two parameter choice rules

In this section, we choose a regularization parameter $\textrm {k}:= \textrm {k}(\upepsilon )$ such that $\| \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \|_{L^{2}(\boldsymbol {\varOmega })} \to 0$ as $\upepsilon \to 0$, and we also consider the convergence analysis between the regularized solution $\mathfrak {u}_{\textrm {k}, \upalpha } ^{\upepsilon }$ and the exact solution $\mathfrak {u}$.

4.1 The a priori parameter choice

Theorem 4.1

Let $\textrm {h}\in L^{2}(\boldsymbol {\varOmega })$and $\textrm {F}\in L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))$. Assume the a priori bound condition (22) holds. If we choose the regularization parameter

$$\begin{aligned} \textrm {k}= \biggl\lfloor \biggl( \frac{\textbf{E}}{\upepsilon } \biggr)^{ \frac{2}{\textrm {r}+1}} \biggr\rfloor , \end{aligned}$$

then we get the following error estimate between the exact solution and its regularization solution with noisy data:

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{- \frac{\textrm {r}}{2}} \biggl( \frac{\textrm {r}}{2} \biggr)^{\frac{\textrm {r}}{2}} \textbf{E}^{\frac{1}{\textrm {r}+ 1}} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+ 1}} + \upmu ^{ \frac{1}{2}} ( 1 + \mathcal{R} ) \textbf{E}^{ \frac{1}{\textrm {r}+ 1}} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+ 1}} + \upepsilon \mathcal{R}, \end{aligned}$$

where $\lfloor \textrm {k}\rfloor$denotes the largest integer less than or equal to k.

Proof

From the triangle inequality, we obtain

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha } - \mathfrak {u}_{\textrm {k}, \upalpha } ^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

(38)

Apply part (a) of Theorem 3.1 and we get

$$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha } - \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} & \le \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha } - \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L ^{2}(\boldsymbol {\varOmega }))} \\ &\le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \bigl\Vert \textrm {h}- \textrm {h}^{\upepsilon } \bigr\Vert _{L ^{2}(\boldsymbol {\varOmega })} + \bigl( \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} + 1 \bigr) \mathcal{R} \bigl\Vert \textrm {F}- \textrm {F}^{\upepsilon } \bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \\ &\le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \upepsilon ( 1 + \mathcal{R} ) + \upepsilon \mathcal{R}. \end{aligned}$$

(39)

On the other hand, we have

$$\begin{aligned} \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} = \bigl\Vert \bigl( \mathcal {A}_{\upgamma } ^{1} (\textrm {t}, \mathscr{T}_{o}) - \mathcal {A}_{\upgamma }^{3} (\textrm {t}, \mathscr{T}_{o}) \bigr) \boldsymbol {\varUpsilon }\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Note $\upalpha \in ( \frac{1}{2},1]$ and $0 < \upmu < \frac{1}{ \Vert \mathcal {K}\Vert ^{2}}$, so it follows that

$$\begin{aligned} & \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \\ &\qquad{} \times \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}} _{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} . \end{aligned}$$

From the definition of ϒ in (17), we get

$$\begin{aligned} \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })}\le{}& \Biggl\Vert \sum_{\textrm {m}=1} ^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}} _{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ \le{}& \upmu ^{-\frac{\textrm {r}}{2}}\sup_{\widetilde{\textrm{a}}_{\textrm {m}}>0} \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggl( \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr)^{\frac{\textrm {r}}{2}} \\ &{} \times \sqrt{\sum_{\textrm {m}=1}^{\infty }\exp \biggl( 2\textrm {r}\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}}. \end{aligned}$$

Apply Lemma 2.1 and we obtain

$$\begin{aligned} \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{-\frac{\textrm {r}}{2}} \biggl( \frac{\textrm {r}}{2} \biggr)^{\frac{\textrm {r}}{2}} \textrm {k}^{-\frac{\textrm {r}}{2}} \textbf{E}. \end{aligned}$$

(40)

Combining (38), (39) and (40) we obtain

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{- \frac{\textrm {r}}{2}} \biggl( \frac{\textrm {r}}{2} \biggr)^{\frac{\textrm {r}}{2}} \textrm {k}^{-\frac{\textrm {r}}{2}}\textbf{E} + \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \upepsilon ( 1 + \mathcal{R} ) + \upepsilon \mathcal{R}. \end{aligned}$$

Choosing the regularization parameter k,

$$\begin{aligned} \textrm {k}= \biggl\lfloor \biggl( \frac{\textbf{E}}{\upepsilon } \biggr)^{ \frac{2}{\textrm {r}+1}} \biggr\rfloor , \end{aligned}$$

we obtain the error estimate

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \upmu ^{- \frac{\textrm {r}}{2}} \biggl( \frac{\textrm {r}}{2} \biggr)^{\frac{\textrm {r}}{2}} \textbf{E}^{\frac{1}{\textrm {r}+ 1}} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+ 1}} + \upmu ^{ \frac{1}{2}} ( 1 + \mathcal{R} ) \textbf{E}^{ \frac{1}{\textrm {r}+ 1}} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+ 1}} + \upepsilon \mathcal{R}. \end{aligned}$$

 □

4.2 A posteriori parameter choice rule and convergence estimate

In the above result, we obtained an error estimate between the exact solution and its regularization solution with noisy data by choosing the a priori parameter k, and this k depends on the noise level ε and the a priori bound condition E. Now, from results in Morozov’s discrepancy principal [11], we choose the regularization parameter k by using an a posteriori choice rule.

The general a posteriori rule can be formulated as follows:

$$\begin{aligned} \bigl\Vert \mathcal {K}\mathfrak {g}_{\textrm {k}, \upalpha }^{\upepsilon } - \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \chi \upepsilon \le \bigl\Vert \mathcal {K}\mathfrak {g}_{\textrm {k}-1, \upalpha }^{\upepsilon } - \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}, \end{aligned}$$

(41)

where $\| \boldsymbol {\varUpsilon }^{\upepsilon } \|_{L^{2}(\boldsymbol {\varOmega })} \ge \chi \upepsilon$, χ is a constant independent of ε and $\textrm {k}>0$ is the regularization parameter which makes (41) hold at the first iteration time.

Choosing $\chi > 1$ the following lemma gives a bound for k in terms of ε and E.

Lemma 4.1

Let $\chi > 1$and k satisfies (41). Also assume the a priori bound condition of $\mathfrak {g}$satisfies (22). Then

$$\begin{aligned} \textrm {k}\le \frac{\textrm {r}-1 }{2 \upmu } \biggl( \frac{1}{\chi - \mathcal {R}-1} \biggr)^{\frac{2}{\textrm {r}-1 }} \biggl( \frac{\textbf{E}}{\upepsilon } \biggr)^{\frac{2}{\textrm {r}-1 }} . \end{aligned}$$

Proof

From the definition of k we get

$$\begin{aligned} & \bigl\Vert \mathcal {K}\mathfrak {g}_{\textrm {k}-1, \upalpha }^{\upepsilon } - \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl(1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o} ^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \biggr]^{\upalpha } \biggr) \bigl\langle \boldsymbol {\varUpsilon }^{\upepsilon }, \textrm{e}_{\textrm {m}} \bigr\rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl(1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o} ^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \biggr]^{\upalpha } \biggr) \bigl\langle \boldsymbol {\varUpsilon }^{\upepsilon }-\boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \bigr\rangle \textrm{e} _{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\qquad {} + \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl(1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o} ^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \biggr]^{\upalpha } \biggr) \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Since $\upalpha \in ( \frac{1}{2},1]$ and $0 < \upmu < \frac{1}{ \Vert \mathcal {K}\Vert ^{2}}$ it follows that

$$\begin{aligned} & \bigl\Vert \mathcal {K}\mathfrak {g}_{\textrm {k}-1, \upalpha }^{\upepsilon } - \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \bigl\Vert \boldsymbol {\varUpsilon }^{\upepsilon }-\boldsymbol {\varUpsilon }\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Apply Lemma 2.1 and we obtain

$$\begin{aligned} & \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \upmu ^{-\frac{1+\textrm {r}}{2}}\sup_{\widetilde{\textrm{a}}_{\textrm {m}}>0} \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}-1} \biggl( \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o} ^{\upgamma }}{\upgamma } \biggr) \biggr)^{\frac{\textrm {r}+1}{2}} \\ & \qquad {}\times \sqrt{\sum_{\textrm {m}=1}^{\infty }\exp \biggl( 2\textrm {r}\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2}} \\ &\quad \le \upmu ^{-\frac{1+\textrm {r}}{2}} \biggl( \frac{\textrm {r}+1}{2} \biggr)^{ \frac{\textrm {r}+1}{2}} \textrm {k}^{\frac{\textrm {r}+1}{2}} \textbf{E}. \end{aligned}$$

Apply Lemma 2.4 and we obtain

$$\begin{aligned} \bigl\Vert \mathcal {K}\mathfrak {g}_{\textrm {k}-1, \upalpha }^{\upepsilon } - \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le{}& \bigl\Vert \textrm {h}^{\upepsilon }- \textrm {h}\bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} + \mathcal{R} \bigl\Vert \textrm {F}^{\upepsilon } - \textrm {F}\bigr\Vert _{L^{\infty }(0,\mathscr {T}_{o};L^{2}(\boldsymbol {\varOmega }))} \\ &{}+ \upmu ^{-\frac{1+\textrm {r}}{2}} \biggl( \frac{\textrm {r}+1}{2} \biggr)^{ \frac{\textrm {r}+1}{2}} \textrm {k}^{-\frac{\textrm {r}+1}{2}} \textbf{E}. \end{aligned}$$

This implies that

$$\begin{aligned} \chi \upepsilon \le ( 1 + \mathcal {R})\upepsilon + \upmu ^{-\frac{\textrm {r}+1}{2}} \biggl( \frac{\textrm {r}+1}{2} \biggr)^{\frac{\textrm {r}+1}{2}} \textrm {k}^{-\frac{\textrm {r}+1}{2}} \textbf{E}, \end{aligned}$$

so

$$\begin{aligned} \textrm {k}\le \frac{\textrm {r}+1 }{2 \upmu } \biggl( \frac{1}{\chi - \mathcal {R}-1} \biggr)^{\frac{2}{\textrm {r}+1 }} \biggl( \frac{\textbf{E}}{\upepsilon } \biggr)^{\frac{2}{\textrm {r}+1 }} . \end{aligned}$$

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Theorem 4.2

Let k be as Lemma 4.1. Assume the a priori bound condition (22) holds. Then we get

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}\le{}& \upmu ^{ \frac{1}{2}} ( 1 + \mathcal{R} ) \biggl( \frac{\textrm {r}+1 }{2 \upmu } \biggr)^{\frac{1}{2}} \biggl( \frac{1}{\chi - \mathcal {R}-1} \biggr)^{\frac{1}{\textrm {r}+ 1 }} \mbox{\textbf{E}}^{\frac{1}{\textrm {r}+ 1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} \\ &{} + \upepsilon \mathcal{R} + ( 1 + \mathcal {R}+ \chi )^{\frac{\textrm {r}}{\textrm {r}+1 }} \mbox{\textbf{E}}^{\frac{1}{\textrm {r}+1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} . \end{aligned}$$

Proof

Using the triangle inequality, we obtain

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} \le \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} + \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha } - \mathfrak {u}_{\textrm {k}, \upalpha } ^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}. \end{aligned}$$

Apply the result of Theorem 4.1 and we obtain

$$\begin{aligned} \bigl\Vert \mathfrak {u}_{\textrm {k}, \upalpha } - \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })} & \le \upmu ^{\frac{1}{2}} \textrm {k}^{\frac{1}{2}} \upepsilon ( 1 + \mathcal{R} ) + \upepsilon \mathcal{R} \\ &\le \upmu ^{\frac{1}{2}} ( 1 + \mathcal{R} ) \biggl( \frac{\textrm {r}+1 }{2 \upmu } \biggr)^{\frac{1}{2}} \biggl( \frac{1}{\chi - \mathcal {R}-1} \biggr)^{\frac{1}{\textrm {r}+ 1 }} \textbf{E}^{\frac{1}{\textrm {r}+ 1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} + \upepsilon \mathcal{R} . \end{aligned}$$

(42)

Apply Hölder’s inequality and we have

$$\begin{aligned} &\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad = \Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( - \widetilde{\textrm{a}}_{\textrm {m}}\frac{ \textrm {t}^{\upgamma } - \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \\ &\qquad {}\times\langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T} _{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{\frac{\textrm {r}}{\textrm {r}+1 }} \\ & \qquad {}\times \Biggl\Vert \sum_{\textrm {m}=1}^{\infty } \exp \biggl( \textrm {r}\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{\frac{1}{\textrm {r}+1 }}. \end{aligned}$$

On the other hand, from $\upalpha \in ( \frac{1}{2},1]$ and $0 < \upmu < \frac{1}{ \Vert \mathcal {K}\Vert ^{2}}$ we get

$$\begin{aligned} &\Biggl\Vert \sum_{\textrm {m}=1}^{\infty }\exp \biggl( \textrm {r}\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma } }{\upgamma } \biggr) \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2 \widetilde{ \textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \textrm{e}_{\textrm {m}} \Biggr\Vert _{L^{2}(\boldsymbol {\varOmega })} ^{\frac{1}{\textrm {r}+1 }} \\ &\quad \le \Biggl( \sum_{\textrm {m}=1}^{\infty }\exp \biggl( 2\textrm {r}\widetilde{\textrm{a}}_{\textrm {m}}\frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \bigl\vert \langle \mathfrak {g}, \textrm{e}_{\textrm {m}} \rangle \bigr\vert ^{2} \Biggr)^{\frac{1}{2(\textrm {r}+1) }} \le \textbf{E}^{\frac{1}{\textrm {r}+1 }}. \end{aligned}$$

This implies that

$$\begin{aligned} & \Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha } \Vert _{L^{2}(\boldsymbol {\varOmega })} \\ &\quad \le \textbf{E}^{\frac{1}{\textrm {r}+1 }} \Biggl( \bigl\vert \boldsymbol {\varUpsilon }- \boldsymbol {\varUpsilon }^{\upepsilon } \bigr\vert _{L^{2}(\boldsymbol {\varOmega })} \\ & \qquad {}+ \Bigg\| \sum_{\textrm {m}=1}^{\infty } \biggl( 1 - \biggl[ 1- \biggl( 1 - \upmu \exp \biggl( -2\widetilde{\textrm{a}}_{\textrm {m}} \frac{ \mathscr{T}_{o}^{\upgamma }}{\upgamma } \biggr) \biggr) ^{\textrm {k}} \biggr]^{\upalpha } \biggr) \langle \boldsymbol {\varUpsilon }, \textrm{e}_{\textrm {m}} \rangle \textrm{e} _{\textrm {m}}\Bigg\| _{L^{2}(\boldsymbol {\varOmega })} \Biggr)^{\frac{\textrm {r}}{\textrm {r}+1 }} \\ &\quad \le ( 1 + \mathcal {R}+ \chi )^{\frac{\textrm {r}}{\textrm {r}+1 }} \textbf{E}^{\frac{1}{\textrm {r}+1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} . \end{aligned}$$

From the above we deduce that

$$\begin{aligned} \bigl\Vert \mathfrak {u}- \mathfrak {u}_{\textrm {k}, \upalpha }^{\upepsilon } \bigr\Vert _{L^{2}(\boldsymbol {\varOmega })}\le{}& \upmu ^{ \frac{1}{2}} ( 1 + \mathcal{R} ) \biggl( \frac{\textrm {r}+1 }{2 \upmu } \biggr)^{\frac{1}{2}} \biggl( \frac{1}{\chi - \mathcal {R}-1} \biggr)^{\frac{1}{\textrm {r}+ 1 }} \textbf{E}^{\frac{1}{\textrm {r}+ 1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} \\ &{} + \upepsilon \mathcal{R} + ( 1 + \mathcal {R}+ \chi )^{\frac{\textrm {r}}{\textrm {r}+1 }} \textbf{E}^{\frac{1}{\textrm {r}+1 }} \upepsilon ^{\frac{\textrm {r}}{\textrm {r}+1 }} . \end{aligned}$$

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