Well-posed results for nonlocal biparabolic equation with linear and nonlinear source terms

In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while we apply the Banach fixed-point theorem to study the existence and uniqueness of the mild solution for the nonlinear source term. In both cases, we show that the mild solution of our problem converges to the solution of an initial value problem as the parameter epsilon tends to zero. The novelty in our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.


Introduction
Let be a bounded domain in R N (N ≥ 1) with sufficiently smooth boundary ∂ . In this paper, we consider the following biparabolic equation: Here u(x, t) is a function of temperature or concentration, F(u) is a source function, ε is a parameter, and f ∈ L 2 ( ) ∩ H s ( ). When ε = 0, the problem becomes an initial conditional problem.
The main equation of problem (1.1) is equivalent to where P is the main operator for the classical parabolic equation, As mentioned by Fushchich, Galitsyn, and Polubinskii [9], the new fourth-order partial differential equation (1.1) is invariant with respect to the Galilei group. From the results in [9] we realize that the classical heat equations do not completely describe heat and mass transfer processes. Therefore, in many situations of heat conduction, it tends to replace the classical thermal equation by one of the hyperbolic form, such as problem (1.1). Problem (1.1) is a form of quadratic PDEs equations, which have a wide range of applications in various scientific and engineering disciplines, such as conduction of heat [7,9,24,33], dynamics of filtration consolidation [6,8], strongly damped wave equations [14,23,34], ice formulation and accretion problems on structures, ships, and aircraft [19][20][21], the transport of liquids and insoluble surfactant through the lung airways [11,12], brain imaging for the detection and mapping of subtle abnormalities of shape and volume in the brains of patients with metastatic tumors [18,26,27], and so on. Whereas there were a number of studies focused on parabolic equations [1-4, 10, 13, 15, 22, 25, 28], the studies on biparabolic equations are still limited. Let us mention previous works related to biparabolic equation (1.1). Lakhdari and Boussetila [16] applied Kozlov-Maz'ya iteration method for approximating the final value problem for biparabolic equation. Bulavatsky [7] studied some boundary value problems for biparabolic equations with nonlocal boundary conditions. Besma et al. [5] considered the problem of approximating a solution of an ill-posed biparabolic problem in the abstract Hilbert space. They introduced a modified quasi-boundary value method to get stable solutions for regularizing the ill-posedness of a biparabolic equation. Tuan et al. [32] studied the problem of finding the initial distribution for a linear inhomogeneous or nonlinear biparabolic equation. Recently, Phuong et al. [25] studied an inverse source problem of the biparabolic equation. Very recently, Tuan et al. [31] investigated two terminal value problems for stochastic biparabolic equations perturbed by a standard Brownian motion or a fractional Brownian motion.
The nonlocal problem focused in this paper is considered as one of the most interesting areas for the readers in various applications, such as chaos, chemistry, biology, and physics; see [30]. In comparison with the initial or final conditions, the nonlocal conditions are more difficult to handle. The novelty of our problem is the presence of condition of nonlocal type (1.2). In many real-world applications, it is difficult to collect accurate data at the beginning or at the end of a process. In addition, many processes happen so fast and in a short period, in which we only can observe the data at the beginning and the end of a process, not the data at a specific time in the range of (0,T). Therefore studies on nonlocal conditional problems can help us to track down a process in more detail and in an effective way.
To the best of our knowledge, up to date, there is still no any study considering problem (1.1) under the nonlocal condition (1.2). This motivates us to focus on problems (1.1)-(1.2). The main contributions of the paper are as follows.
• For the linear source function, we give the well-posedness and investigate the convergence of the mild solution to problem (1.1)-(1.2) as ε approaches 0. In more detail, we prove that the solution of problem (1.1)-(1.2) converges to a mild solution with the initial value problem for (1.1). • For nonlinear source functions, we prove the existence and uniqueness of mild solutions. In the main analysis, we apply the Banach fixed point theorem. Our next aim is to demonstrate the convergence of the mild solution as the parameter ε tends to 0. The main techniques to handle the above problem are based on the ideas of some recent publications [17,29,30]. We overcome some difficulties by setting up complex evaluations on Hilbert scale spaces. Choosing the right spaces for the input f and for the solution is also not simple task.
This paper is organized as follows. In Sect. 2, we provide some useful notations and the definition of a solution in the mild sense. In Sect. 3, we focus on the well-posed results for the linear case and discuss on what happens as ε → 0. The well-posed results for the nonlinear source term are introduced in Sect. 4. Eventually, the results are summarized in Sect. 5.

Preliminary results and mild solution
In this section, we introduce the notation and the functional setting used in our paper. Recall that the spectral problem admits eigenvalues 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n ≤ · · · with λ n → ∞ as n → ∞. The corresponding eigenfunctions are ψ n ∈ H 1 0 ( ).

Definition 2.1 (Hilbert scale space)
We recall the Hilbert scale space given as follows: Let us give an explicit formula of the mild solution. First, taking the inner product of both sides of (1.1) with ψ n (x), we find that It is easy to see that the latter problem has a solution given by We rewrite it as Combining (2.2) and (2.4), we find that and From (2.5) we give an explicit formula of the solution to problem (1.1)-(1.2) in the mild setting:

Well-posed results for linear case
In this section, we focus on the case F(t, u) = F(t). Under the linear case, we recall the mild solution u ε to problem (1.1)-(1.2): Proof Using Parseval's equality, we find that In view of the inequality e -z ≤ C ν z -ν for all ν > 0, we know that It follows from (3.4) that which gives the estimate Setting ν = m + 1s > 0, we know that Using again e -z ≤ C ν z -ν for all ν > 0, we find that Proof Applying Lemma 3.1 and noting that m < s < m + 1, we find that Let μ be such that 1 < μ < 1 m+1-s . The first term I 1 is bounded by (3.12) For the second term I 2 , we easily observe that Then we get the bound (3.14) For the third term I 3 , using Lemma (3.1), we have that This immediately implies that Combining (3.11), (3.12), (3.14), and (3.16), we find that Since (3.1), we get that (3.20) Our next aim is estimating the term (Q ε (t) -S(t))f . We clearly see that Parseval's equality implies that (3.21) Using the inequality e -z ≤ C ν z -ν for all ν > 0, we arrive at e -2tλ n ≤ C(m, s)(tλ n ) -2(m+1-s) . (3.22) It is obvious that It follows from (3.21) that This implies that where we recall that 1 < μ < 1 m+1-s . Combining (3.19), (3.20), and (3.24), we arrive at for all ϕ, ψ ∈ H p ( ) and p < q < p + 1. Then for any ε > 0 and K f small enough, problem (1.1)-(1.2) has a unique mild solution in X a,∞ ((0, T]; H p ( )), which satisfies where In addition, Proof We look for the solution in the space X a,∞ ((0, T]; H p ( )). Let us define the function If ψ = 0, then by the assumption F(0) = 0 we have that Since s < p + 1, we set ν = p + 1s. Then it follows from (4.6) that Under the assumption p + 1 ≤ s + a, if ψ = 0, then we find that for any 0 ≤ t ≤ T, which allows us to derive that B ε ψ belongs to the space X a,∞ ((0, T]; H p ( )) if ψ = 0. Let ϕ, ψ ∈ X a,∞ ((0, T]; H p ( )). It is obvious that By the second part of Lemma 3.1 the term J 1 is bounded by where we note that p > q. Since F is globally Lipschitz as in (4.1), we infer that where we note that q + 2 > p and a < 1. This implies that The right-hand side of this expression is independent of t, and we deduce that Since q < p + 1 and a > 0, we can choose a real number s such that max(p + 1a, q) ≤ s < p + 1.

Theorem 4.2 Let F be as in
Proof Let us recall that u(t) = S(t)f + t 0 (tr)S(tr)F u(r) dr, (4.25) where we recall that By (4.2) we immediately have the result on the difference between u ε (t) and u(t) which is split as the sum of three terms u ε (t)u(t) = Q ε (t)f -S(t)f + for the linear source term and applied the Banach fixed point theorem to study the existence and uniqueness of a mild solution for the nonlinear source term. In both cases, we demonstrated that the mild solution of our problem converges to the solution of an initial value problem as the parameter ε → 0. The most compelling findings of our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.