Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

Abstract

We prove the global existence of small data solution in all spaces of all dimensions $n\geq 1$ for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms $f(t,u)$ and $g(t,v)$ satisfy some properties of parabolic equations. We study the problem in several classes of regularity.

1 Introduction

Let us consider the Cauchy problem for the semilinear classical damped wave equation with power nonlinearity

$$ u_{tt}-\Delta u+u_{t}=f(u),\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x), $$

(1)

where $t\in [0,\infty ), x\in \mathbb{R}^{n}$, and

$$ f(0)=0, \qquad\bigl\vert f(u)-f(\tilde{u}) \bigr\vert \lesssim \vert u-\tilde{u} \vert \bigl( \vert u \vert + \vert \tilde{u} \vert \bigr)^{p-1}. $$

(2)

Having the estimates proved in [17] for the corresponding homogeneous problem, for given compactly supported initial data $(u_{0},u_{1}) \in H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ and for $p\leq p_{GN}(n):=\frac{n}{n-2}$ if $n\geq 3$, the authors in [22] proved the local (in time) existence of energy solutions $u\in \mathcal{C}([0,T), H^{1}(\mathbb{R}^{n}))\cap \mathcal{C}^{1}([0,T), L^{2}(\mathbb{R}^{n}))$. Moreover, they proved the global (in time) existence of small data solutions by using the technique of “potential well” and “modified potential well”. The Cauchy problem (1) was also studied in [7, 12, 13, 27, 30], where the Fujita exponent $p_{Fuj}(n):=1+\frac{2}{n}$ has an important role as the critical exponent. The critical exponent means that we have the global (in time) existence of small data weak solutions for $p>p_{Fuj}(n)$, whereas the local (in time) existence for $p>1$ and large data can be only expected.

Assuming a time-dependent coefficient in the dissipation term, we first consider the Cauchy problem

$$ u_{tt}-\Delta u+b(t)u_{t}=0,\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x). $$

Among other classifications of the dissipation term $b(t)u_{t}$ introduced in [28] and [29], we are interested in the effective case, where $b=b(t)$ satisfies the following properties:

• b is a positive and monotonic function with $t b(t)\rightarrow \infty$ as $t\rightarrow \infty$,

• $((1+t)^{2}b(t))^{-1} \in L^{1}(0,\infty )$,

• $b\in \mathcal{C}^{3}[0,\infty )$ and $|b^{(k)}(t)| \lesssim \frac{b(t)}{(1+t)^{k}}$ for $k=1,2,3$,

• $\frac{1}{b} \notin L^{1}(0,\infty )$, and there exists a constant $a\in [0,1)$ such that $tb'(t)\leq ab(t)$.

Examples of functions belonging to this class are the followings with $r\in (-1,1)$:

• $b(t)=\frac{\mu }{(1+t)^{r}}$ for some $\mu >0$, $b(t)=\frac{\mu }{(1+t)^{r}}(\log (e+t))^{\gamma }$ for some $\mu >0$ and $\gamma >0$, and $b(t)=\frac{\mu }{(1+t)^{r}(\log (e+t))^{\gamma }}$ for some $\mu >0$ and $\gamma >0$.

In [5] the authors derived such estimates for solutions to the family of parameter-dependent Cauchy problems

$$ u_{tt}-\Delta u+b(t)u_{t}=0,\qquad v(\tau,x)=0,\qquad v_{t}(\tau,x)=f(u) ( \tau,x). $$

Using theses estimates together with Duhamel’s principle, in the same paper the authors proved the global existence of small data solutions to the following semilinear Cauchy problem:

$$ u_{tt}-\Delta u+b(t)u_{t}=f(u), \qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x), $$

where $f(u)$ satisfies condition (2).

In 2013, D’Abbicco [3] proved the global existence of small data solution for low space dimensions and derived decay estimates for solutions to the Cauchy problem

$$ u_{tt}-\Delta u+b (t)u_{t}=f(t,u),\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x), $$

where

$$ f(t,0)=0 \quad\text{and}\quad \bigl\vert f(t,v)-f(t,\tilde{v}) \bigr\vert \lesssim \biggl(1+ \int _{0}^{t}\frac{1}{b(r)}\,dr \biggr)^{\gamma } \vert v-\tilde{v} \vert \bigl( \vert v \vert + \vert \tilde{v} \vert \bigr)^{p-1}. $$

Weakly coupled systems can be an interesting problem, treated and improved in [16] and [1]. In this paper, we study in all space dimensions the Cauchy problem of weakly coupled system of semilinear effectively damped waves

$$ \begin{aligned} &u_{tt}-\Delta u+b_{1}(t)u_{t}=f(t,v),\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x), \\ &v_{tt}-\Delta v+b_{2}(t)v_{t}= g(t,u),\qquad v(0,x)=v_{0}(x),\qquad v_{t}(0,x)=v_{1}(x), \end{aligned} $$

(3)

where

$$\begin{aligned} &\bigl(1+B_{1}(t,0)\bigr)^{\beta } \lesssim \bigl(1+B_{2}(t,0)\bigr)\lesssim \bigl(1+B_{1}(t,0) \bigr)^{ \frac{1}{\alpha }}, \end{aligned}$$

(4)

$$\begin{aligned} &f(t,0)=0, \qquad\bigl\vert f(t,v)-f(t,\tilde{v}) \bigr\vert \lesssim \bigl(1+B_{1}(t,0)\bigr)^{ \gamma _{1}} \vert v-\tilde{v} \vert \bigl( \vert v \vert + \vert \tilde{v} \vert \bigr)^{p-1}, \end{aligned}$$

(5)

$$\begin{aligned} &g(t,0)=0,\qquad \bigl\vert g(t,u)-g(t,\tilde{u}) \bigr\vert \lesssim \bigl(1+B_{2}(t,0)\bigr)^{ \gamma _{2}} \vert u-\tilde{u} \vert \bigl( \vert u \vert + \vert \tilde{u} \vert \bigr)^{q-1}, \end{aligned}$$

(6)

for $B_{1}(t,\tau )=\int _{\tau }^{t}\frac{1}{b_{1}(r)}\,dr, B_{2}(t,\tau )= \int _{\tau }^{t}\frac{1}{b_{2}(r)}\,dr, \alpha,\beta \in \mathbb{R}_{+}^{*}$, and $\gamma _{1},\gamma _{2}\in [-1,\infty )$. If we take $\gamma _{1}<-1$ or $\gamma _{2}<-1$, then we will get an empty admissible range for p or q (see the table in Remark 2.3).

Recently, Nishihara and Wakasugi [23] studied the particular case of (3), where $b_{1}(t)=b_{2}(t)=1,f(t,v)=|v|^{p}$, and $g(t,u)=|u|^{q}$. Using the weighted energy method, they proved the global (in time) existence if the inequality

$$ \frac{\max \{p;q\}+1}{pq-1}< \frac{n}{2}$$

(7)

is satisfied. Using an additional regularity $L^{m}(\mathbb{R}^{n})$ for data, we conclude the so-called modified Fujita exponent $p_{Fuj,m }:=1+\frac{2m}{n}$; this new exponent implies a modified condition corresponding to (7), $\frac{\max \{p;q\}+1}{pq-1}<\frac{n}{2m}$. In [20] and [18] the authors studied the above system with the same nonlinearities assumed in [23] by taking the equivalent coefficients $b_{1}=b_{1}(t)$ and $b_{2}=b_{2}(t)$ or, in other words, $\alpha =\beta =1$. The global (in time) existence of small initial data solutions was proved assuming different classes of regularity of data and for all space dimensions. Considering (3) in [21], the authors proved a global existence result for a particular case from the set of effective dissipation terms $b_{1}(t)=\frac{\mu }{(1+t)^{r_{1}}}, r_{1},r_{2}\in (-1,1)$, and $b_{2}(t)=\frac{\mu }{(1+t)^{r_{2}}}$ with the nonlinearities $f(t,v)=|v|^{p}$ and $f(t,u)=|u|^{q}$.

1.1 Notations

For $s>0$ and $m \in [1,2)$, we introduce the function space

$$ \mathcal{A}_{m,s}:=\bigl(H^{s}\bigl(\mathbb{R}^{n} \bigr)\cap L^{m}\bigl(\mathbb{R}^{n}\bigr)\bigr) \times \bigl(H^{s-1}\bigl(\mathbb{R}^{n}\bigr)\cap L^{m} \bigl(\mathbb{R}^{n}\bigr)\bigr) $$

with the norm

$$ \bigl\Vert (u,v) \bigr\Vert _{\mathcal{A}_{m,s}}:= \Vert u \Vert _{H^{s}}+ \Vert u \Vert _{L^{m}}+ \Vert v \Vert _{H^{s-1}}+ \Vert v \Vert _{L^{m}}. $$

We denote by p̃ and q̃ the modified exponents of the exponents p and q in the power nonlinearities appearing in (5) and (6). Then

$$ \tilde{p}= \textstyle\begin{cases} (p-1)\beta +1 & \text{if } \beta \geq 1, \\ (p-\frac{m}{2})\beta +\frac{m}{2} &\text{if } 0< \beta < 1, \end{cases} $$

(8)

and

$$ \tilde{q}= \textstyle\begin{cases} (q-1)\alpha +1 & \text{if } \alpha \geq 1, \\ (q-\frac{m}{2})\alpha +\frac{m}{2} & \text{if } 0< \alpha < 1. \end{cases} $$

(9)

Remark 1.1

If $\alpha =\beta =1$, then $(1+B_{1}(t,0)) \approx (1+B_{2}(t,0))$. This case was studied in previous papers. In this work, we restrict ourselves to the remaining cases.

2 Main results

We study the Cauchy problem (3) in several cases with respect to the regularity of the data to cover all space dimensions and the modified exponents of power nonlinearities $\tilde{p}, \tilde{q}$ and parameters $\alpha,\beta,\gamma _{1},\gamma _{2}$. Therefore we introduce the following classification of regularity: Data from energy space $s=1$, data from Sobolev spaces with suitable regularity $s\in (1,\frac{n}{2}+1]$, and, finally, large regular data $s>\frac{n}{2}+1$.

2.1 Data from the energy space

In this section, we are interested in system (3), where the data are taken from the function space $\mathcal{A}_{m,1}$. In Theorem 2.1, we treat the case where both modified exponents power p̃ and q̃ are above the modified Fujita exponents

$$ p_{Fuj,m,\gamma _{1} }:=1+\frac{2m(\gamma _{1} +1)}{n} \quad\text{and}\quad q_{Fuj,m,\gamma _{2} }:=1+ \frac{2m(\gamma _{2} +1)}{n}, $$

respectively.

Theorem 2.1

Let the data $(u_{0},u_{1}),(v_{0},v_{1})$ belong to $\mathcal{A}_{m,1} \times \mathcal{A}_{m,1}$ for $m\in [1,2)$. Moreover, let the modified exponents satisfy

$$\begin{aligned} \tilde{p}>p_{Fuj,m,\gamma _{1} },\qquad \tilde{q}>p_{Fuj,m,\gamma _{2} }, \end{aligned}$$

(10)

and let the exponents p and q of the power nonlinearities satisfy

$$ \begin{aligned}& \frac{2}{m}\leq \min \{p;q\}\leq \max \{p;q\} < \infty \quad \textit{if } n \leq 2, \\ &\frac{2}{m}\leq \min \{p;q\}\leq \max \{p;q\}\leq p_{GN}(n)\quad \textit{if } n>2. \end{aligned} $$

(11)

Then there exists a constant $\epsilon _{0}$ such that if

$$ \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \leq \epsilon _{0}, $$

then there exists a uniquely determined global (in time) energy solution to (3) in

$$ \bigl(\mathcal{C} \bigl([0,\infty), H^{1}\bigl( \mathbb{R}^{n}\bigr) \bigr)\cap \mathcal{C}^{1} \bigl([0, \infty), L^{2}\bigl(\mathbb{R}^{n}\bigr)\bigr) \bigr)^{2}. $$

Furthermore, the solution satisfies the following decay estimates:

$$\begin{aligned} \bigl\Vert \nabla ^{j} \partial _{t}^{l}u(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim{}& b_{1}(t)^{-l} \bigl(1+B_{1}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{j}{2} -l} \\ &{} \times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \bigr), \\ \bigl\Vert \nabla ^{j} \partial _{t}^{l}v(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim{}& b_{2}(t)^{-l} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{j}{2} -l} \\ &{} \times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \bigr), \end{aligned}$$

where $j+l=0,1$.

Remark 2.2

We remark that for $\gamma _{1}=\gamma _{2}=0$, system (3) behaves in this case like one single equation because the modified power nonlinearities p̃ and q̃ are influenced separately only by the modified Fujita exponent $p_{Fuj,m}(n)=\frac{2m}{n}+1$. Then we cannot feel in an optimal way the interplay between the powers of nonlinearities in the existence conditions.

Remark 2.3

The final admissible ranges for the exponents p and q of power nonlinearities can be fixed using several parameters such as $\alpha,\beta$, the exponents $\gamma _{1},\gamma _{2}$, the space dimension n, and the parameter of additional regularity m. As an example for the dimension $n=1$, if we take $0<\beta <1$, then $\tilde{p}< p$. We distinguish two cases:

• If $\gamma _{1}\geq -\frac{1}{2}$, then $p\geq \frac{2}{m}$ for $\tilde{p}>p_{Fuj,m,\gamma _{1} }$ which is equivalent to $p>\frac{1}{\beta } (2m(\gamma _{1}+1)-\frac{m}{2} +1 ) + \frac{m}{2}$.

• If $\gamma _{1}\in [-1,-\frac{1}{2})$, then the solution exists for

  $$ p>\max \biggl\lbrace \frac{1}{\beta } \biggl(2m(\gamma _{1}+1)- \frac{m}{2}+1 \biggr)+\frac{m}{2};\frac{2}{m} \biggr\rbrace . $$

The general case for the admissible ranges from below can be summarized as follows:

β	Nonlinearity parameter $\gamma _{1}$	Admissible range for p
0<β<1	$\gamma _{1}\geq -1+\frac{n}{2}$	$p>\frac{1}{\beta }+\frac{2m(\gamma _{1}+1)}{n\beta }-\frac{m}{2\beta }+\frac{m}{2}$
	$\gamma _{1}\in [-1,-1+\frac{n}{2})$	$p>\max \biggl\lbrace \frac{1}{\beta }+\frac{2m(\gamma _{1}+1)}{n\beta }-\frac{m}{2\beta }+ \frac{m}{2}; \frac{2}{m} \biggr\rbrace$
β ≥ 1	$\gamma _{1}\geq -1+\frac{n\beta }{2}$	$p> \frac{2m(\gamma _{1}+1)}{n\beta }+1$
	$\gamma _{1}\in [-1,-1+\frac{n\beta }{2})$	$p> \max \biggl\lbrace \frac{2m(\gamma _{1}+1)}{n\beta }+1;\frac{2}{m} \biggr\rbrace$

In the same way, we can get the admissible range for q with respect to the parameters α and $\gamma _{2}$.

Example 2.4

Let us choose the space dimension $n=2$, the parameters $\gamma _{1}=-1,\gamma _{2}=-\frac{1}{3}$, and the coefficients of the dissipation terms $b_{1}(t)=(1+t)^{ -\frac{1}{2}}$ and $b_{2}(t)=(1+t)^{ \frac{1}{2}}$, which implies $\beta =\frac{1}{\alpha }=3$. Using (10) from the previous theorem for $m=2$, we get $\tilde{p}>1,\tilde{q}>\frac{7}{3}$. Theses conditions together with (11) after applying (8) and (9) imply the following admissible range for the exponents of power nonlinearities:

$$ p>1,\qquad q>5. $$

The case where one exponent p̃ or q̃ is below the modified Fujita exponent, we distinguish four cases with respect to the values of α and β:

• 1. $\tilde{p}\leq 1+\frac{2m(\gamma _{1} +1)}{n},\tilde{q}>1+ \frac{2m(\gamma _{2} +1)}{n}$ with $\min \{\alpha;\beta \}\geq 1$ or $\min \{\alpha;\beta \}\leq 1\leq \max \{\alpha;\beta \}$.

• 2. $\tilde{p}>1+\frac{2m(\gamma _{1} +1)}{n},\tilde{q}\leq 1+ \frac{2m(\gamma _{2} +1)}{n}$ with $\min \{\alpha;\beta \}\geq 1$ or $\min \{\alpha;\beta \}\leq 1\leq \max \{\alpha;\beta \}$.

Theorem 2.5

Let $m\in [1,2), \alpha \geq 1$, and $\beta >0$. The data $(u_{0},u_{1}),(v_{0},v_{1})$ are assumed to belong to $\mathcal{A}_{m,1} \times \mathcal{A}_{m,1}$. Moreover, let the modified exponents satisfy

$$\begin{aligned} \begin{aligned} &\tilde{p}< \frac{2m(\gamma _{1}+1)}{n}+1, \\ &\tilde{q}>\frac{2m(\gamma _{2}+1)}{n}+1. \end{aligned} \end{aligned}$$

(12)

Moreover, we assume that

$$ \frac{n}{2}>m \biggl( \frac{\tilde{q}+\alpha +\gamma _{1}\tilde{q}+\gamma _{1}(\alpha -1) +\gamma _{2}}{ \tilde{p}\tilde{q}-1+(\alpha -1)(\tilde{p}-1)} \biggr) $$

(13)

and the exponents p and q of the power nonlinearities satisfy

$$ \begin{aligned} &\frac{2}{m}\leq \min \{p;q\}\leq \max \{p;q\} < \infty \quad \textit{if } n \leq 2, \\ &\frac{2}{m}\leq \min \{p;q\}\leq \max \{p;q\}\leq p_{GN}(n)\quad \textit{if } n>2. \end{aligned} $$

Then there exists a constant $\epsilon _{0}$ such that if

$$ \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \leq \epsilon _{0}, $$

then there exists a uniquely determined global (in time) energy solution to (3) in

$$ \bigl(\mathcal{C} \bigl([0,\infty), H^{1}\bigl( \mathbb{R}^{n}\bigr) \bigr)\cap \mathcal{C}^{1} \bigl([0, \infty), L^{2}\bigl(\mathbb{R}^{n}\bigr) \bigr) \bigr)^{2}. $$

Furthermore, the solution satisfies the following decay estimates:

$$\begin{aligned} & \bigl\Vert \nabla ^{j} \partial _{t}^{l}u(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &\quad \lesssim b_{1}(t)^{-l} \bigl(1+B_{1}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{j}{2} -l+\kappa ( \tilde{p})} \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \bigr), \\ & \bigl\Vert \nabla ^{j} \partial _{t}^{l}v(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &\quad \lesssim b_{2}(t)^{-l} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{j}{2} -l} \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,1}} \bigr), \end{aligned}$$

where $j+l=0,1$, and

$$ \kappa (\tilde{p}) =\gamma _{1}-\frac{n}{2m}(\tilde{p}-1)+1 $$

represents the loss of decay in comparison with the corresponding decay estimates for the solution u of the linear Cauchy problem with vanishing right-hand side.

Remark 2.6

Choosing $\tilde{p}=p_{Fuj,m}(n)$ in condition (12), we get an arbitrarily small loss of decay $\kappa (\tilde{p})=\varepsilon$.

We summarize the remaining results for all cases with respect to $\alpha,\beta,\tilde{p}$, and q̃ as follows:

• If we assume in the statement of the previous theorem that $\alpha <1$ and $\beta \geq 1$, then, instead of (13), we get the condition

  $$ \frac{n}{2}>m \biggl( \frac{\tilde{q}+1+\gamma _{1}\tilde{q}+\gamma _{2}+\frac{m}{2}(\alpha -1)(\gamma _{1}+1)}{ \tilde{p}\tilde{q}-1+\frac{m}{2}(\alpha -1)(\tilde{p}-1)} \biggr). $$

• If $\tilde{p}>\frac{2m(\gamma _{1}+1)}{n}+1,\tilde{q}\leq \frac{2m(\gamma _{2}+1)}{n}+1$, then, instead of (13), we have to assume that

  $$\begin{aligned} &\frac{n}{2}>m \biggl( \frac{\tilde{p}+\beta +\gamma _{2} \tilde{p}+\gamma _{2}(\beta -1) +\gamma _{1}}{ \tilde{p}\tilde{q}-1+(\beta -1)(\tilde{q}-1)} \biggr) \quad \text{for } \alpha >0, \beta \geq 1, \\ &\frac{n}{2}>m \biggl( \frac{\tilde{p}+1+\gamma _{2} \tilde{p}+\gamma _{1}+\frac{m}{2}(\beta -1)(\gamma _{2}+1)}{ \tilde{p}\tilde{q}-1+\frac{m}{2}(\beta -1)(\tilde{q}-1)} \biggr) \quad \text{for } \alpha \geq 1, \beta < 1. \end{aligned}$$

2.2 Data from Sobolev spaces with suitable regularity

In this section the regularity of data has a strong influence on the admissible range of the modified exponents or the exponents of power nonlinearities, respectively. For this reason, we assume that the data have a different suitable regularity, that is,

$$\begin{aligned} &(u_{0},u_{1})\in H^{s_{1}}\bigl( \mathbb{R}^{n}\bigr)\times H^{s_{1}-1}\bigl( \mathbb{R}^{n} \bigr),\quad s_{1} \in \biggl(1,1+\frac{n}{2} \biggr], \\ &(v_{0},v_{1})\in H^{s_{2}}\bigl( \mathbb{R}^{n}\bigr)\times H^{s_{2}-1}\bigl( \mathbb{R}^{n} \bigr),\quad s_{2} \in \biggl(1,1+\frac{n}{2} \biggr], \end{aligned}$$

with an additional regularity $L^{m}(\mathbb{R}^{n})$, $m \in [1,2)$. In this section, we use a generalized (fractional) Gagliardo–Nirenberg inequality used in [11] and [25]. Furthermore, we use a fractional Leibniz rule and a fractional chain rule, which are explained in the Appendix.

Theorem 2.7

Let $n\geq 4$, $s_{1}\in (3+2\gamma _{1},\frac{n}{2}+1], s_{2}\in (3+2\gamma _{2}, \frac{n}{2}+1]$, $0< s_{2}-s_{1}<1$, and $\lceil s_{1}\rceil \neq \lceil s_{2}\rceil$. The data $(u_{0},u_{1}),(v_{0},v_{1})$ are supposed to belong to $\mathcal{A}_{m,s_{1}}\times \mathcal{A}_{m,s_{2}}$ with $m\in [ 1,2)$. Furthermore, we assume that

$$ \tilde{p}>\frac{2m}{n} \biggl(\frac{s_{1}+1+2\gamma _{1}}{2} \biggr) +1, \qquad\tilde{q}>\frac{2m}{n} \biggl(\frac{s_{2}+1+2\gamma _{2}}{2} \biggr) +1. $$

(14)

and that the exponents p and q of the power nonlinearities satisfy the conditions

$$ \begin{aligned} &\lceil s_{1}\rceil < p,\qquad \lceil s_{2}\rceil < q \quad\textit{if } n\leq 2s_{1}, \\ &\lceil s_{1}\rceil < p,\qquad \lceil s_{2}\rceil < q \leq 1+ \frac{2}{n-2s_{1}} \quad\textit{if } 2s_{1}< n\leq 2s_{2}, \\ &\lceil s_{1}\rceil < p \leq 1+\frac{2}{n-2s_{2}},\qquad \lceil s_{2}\rceil < q \leq 1+\frac{2}{n-2s_{1}} \quad\textit{if } n>2s_{2}. \end{aligned} $$

(15)

Then there exists a constant $\epsilon _{0}$ such that if

$$ \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{ \mathcal{A}_{m,s_{2}}}\leq \epsilon _{0}, $$

then there exists a uniquely determined globally (in time) energy solution to (3) in

$$\begin{aligned} & \bigl(\mathcal{C} \bigl([0,\infty), H^{s_{1}}\bigl( \mathbb{R}^{n}\bigr) \bigr) \cap \mathcal{C}^{1} \bigl([0, \infty), H^{s_{1}-1}\bigl(\mathbb{R}^{n}\bigr)\bigr) \bigr) \\ &\quad{} \times \bigl(\mathcal{C} \bigl([0,\infty), H^{s_{2}}\bigl( \mathbb{R}^{n}\bigr) \bigr)\cap \mathcal{C}^{1} \bigl([0, \infty), H^{s_{2}-1}\bigl( \mathbb{R}^{n}\bigr) \bigr) \bigr). \end{aligned}$$

Furthermore, for $l=0,1$, the solution satisfies the estimates

$$\begin{aligned} \bigl\Vert \vert D \vert ^{s_{1}-l}\partial _{t}^{l} u(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim {}&b_{1}(t)^{-l} \bigl(1+B_{1}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -l-\frac{s_{1}-l}{2}} \\ &{}\times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}} \bigr), \\ \bigl\Vert \vert D \vert ^{s_{2}-l}\partial _{t}^{l}v (t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim {}& b_{2}(t)^{-l} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -l-\frac{s_{2}-l}{2}} \\ &{} \times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}} \bigr). \end{aligned}$$

Particular cases:

• If $\beta \geq 1$ and $s_{1}\geq 3+2\gamma _{1}$, then under the assumptions of Theorem 2.7, the condition $p>\lceil s_{1}\rceil$ implies $\tilde{p}>\frac{2m}{n} (\frac{s_{1}+1+2\gamma _{1}}{2} ) +1$.

• If $\alpha \geq 1$ and $s_{2}\geq 3+2\gamma _{2}$, then under the assumptions of Theorem 2.7, the condition $p>\lceil s_{2}\rceil$ implies $\tilde{q}>\frac{2m}{n} (\frac{s_{2}+1+2\gamma _{2}}{2} ) +1$.

2.3 Large regular data

This case has been classified to benefit from the embedding in $L^{\infty }(\mathbb{R}^{n})$, where the data are supposed to have a high regularity, which means that

$$\begin{aligned} &(u_{0},u_{1})\in H^{s_{1}}\bigl( \mathbb{R}^{n}\bigr)\times H^{s_{1}-1}\bigl( \mathbb{R}^{n} \bigr),\quad s_{1}> \frac{n}{2}+1, \\ &(v_{0},v_{1})\in H^{s_{2}}\bigl( \mathbb{R}^{n}\bigr)\times H^{s_{2}-1}\bigl( \mathbb{R}^{n} \bigr),\quad s_{2}>\frac{n}{2}+1. \end{aligned}$$

Theorem 2.8

Let $n\geq 4$, $(u_{0},u_{1}),(v_{0},v_{1})\in \mathcal{A}_{m,s_{1}}\times \mathcal{A}_{m,s_{2}}$, $m\in [ 1,2)$, $\min \{s_{2}; s_{1}\}>\frac{n}{2}+1$, and $s_{1}-s_{2}\in (-1,1)$. Moreover, let

$$\begin{aligned} p>s_{1},\qquad q>{s}_{2}, \end{aligned}$$

and

$$ \tilde{p}>\frac{2m}{n} \biggl(\frac{s_{1}+1+2\gamma _{1}}{2} \biggr) +1,\qquad \tilde{q}>\frac{2m}{n} \biggl(\frac{s_{2}+1+2\gamma _{2}}{2} \biggr) +1. $$

(16)

Then there exists a constant $\epsilon _{0}$ such that if

$$ \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{ \mathcal{A}_{m,s_{2}}}\leq \epsilon _{0}, $$

then there exists a uniquely determined globally (in time) energy solution to (3) in

$$\begin{aligned} & \bigl(\mathcal{C} \bigl([0,\infty), H^{s_{1}}\bigl( \mathbb{R}^{n}\bigr) \bigr) \cap \mathcal{C}^{1} \bigl([0, \infty), H^{s_{1}-1}\bigl(\mathbb{R}^{n}\bigr) \bigr) \bigr) \\ &\quad{}\times \bigl(\mathcal{C} \bigl([0,\infty), H^{{s}_{2}}\bigl( \mathbb{R}^{n}\bigr) \bigr)\cap \mathcal{C}^{1} \bigl([0,t], H^{{s}_{2}-1}\bigl( \mathbb{R}^{n}\bigr) \bigr) \bigr). \end{aligned}$$

Furthermore, the solution satisfies for $l=0,1$ the estimates

$$\begin{aligned} \bigl\Vert \vert D \vert ^{s_{1}-l}\partial _{t}^{l} u(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim {}&b_{1}(t)^{-l} \bigl(1+B_{1}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -l-\frac{s_{1}-l}{2}} \\ &{} \times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}} \bigr), \\ \bigl\Vert \vert D \vert ^{s_{2}-l}\partial _{t}^{l}v (t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim {}& b_{2}(t)^{-l} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -l-\frac{s_{2}-l}{2}} \\ &{} \times \bigl( \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}} \bigr). \end{aligned}$$

3 Philosophy of our approach and proofs

3.1 Some tools

First, we recall the following result from [5].

Lemma 3.1

The primitive $B=B(t,\tau )$ of $\frac{1}{b}$ satisfies the following properties:

$$\begin{aligned} &B(t,\tau )\approx B(t,0)\quad \textit{for all } \tau \in \biggl[0, \frac{t}{2} \biggr], \end{aligned}$$

(17)

$$\begin{aligned} &B(\tau,0)\approx B(t,0) \quad\textit{for all } \tau \in \biggl[ \frac{t}{2},t \biggr], \\ &\int _{\frac{t}{2}}^{t}\frac{1}{b(\tau )} \bigl(1+B(t,\tau ) \bigr)^{- \frac{j}{2}-l}\,d\tau \lesssim \bigl(1+B(t,0)\bigr)^{1-\frac{j}{2}-l}\log \bigl(1+B(t,0)\bigr)^{l}\quad \textit{for } j+l=0,1. \end{aligned}$$

(18)

To use Duhamel’s principle, we need the following results in the proofs of our main results.

Theorem 3.2

The Sobolev solutions to the Cauchy problem

$$ u_{tt}-\Delta u+b(t)u_{t}=0,\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x) $$

satisfy the following estimates for $t>0$:

For data from the energy space $(s=1)$,

$$ \bigl\Vert \nabla ^{j} \partial _{t}^{l} u(t, \cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(b(t)\bigr)^{-l} \bigl(1+B(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{j}{2} -l} \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,1}}, $$

where $j+l=0,1$;

for high regular data $(s>1)$,

$$\begin{aligned} &\bigl\Vert u(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{ \mathcal{A}_{m,s}}, \\ &\bigl\Vert u_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(t)^{-1} \bigl(1+B(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1} \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s}}, \\ &\bigl\Vert \vert D \vert ^{s} u(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{s}{2}} \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s}}, \\ &\bigl\Vert \vert D \vert ^{s-1} u_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(t)^{-1} \bigl(1+B(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{s-1}{2}-1} \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s}}. \end{aligned}$$

The proof of this theorem follows from [28] and [29].

Theorem 3.3

The Sobolev solutions to the parameter-dependent family of Cauchy problems

$$ v_{tt}-\Delta v+b(t)v_{t}=0,\qquad v(\tau,x)=0,\qquad v_{t}(\tau,x)=v_{1}(x) $$

satisfy the following estimates for $t>\tau, \tau \geq 0$:

For data from the energy space $(s=1)$,

$$\begin{aligned} \bigl\Vert \nabla ^{j}\partial _{t}^{l} v(t,\cdot ) \bigr\Vert _{L^{2}} \lesssim b(t)^{-1}b( \tau )^{-l} \bigl(1+B(t,\tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} ) -\frac{j}{2}-l} \Vert v_{1} \Vert _{L^{2}\cap L^{m}}, \end{aligned}$$

(19)

where $j+l=0, 1$;

For high regular data $(s>1)$,

$$\begin{aligned} &\bigl\Vert v(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(\tau )^{-1} \bigl(1+B(t,\tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \Vert v_{1} \Vert _{H^{s-1}\cap L^{m}}, \\ &\bigl\Vert v_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(\tau )^{-1}b(t)^{-1} \bigl(1+B(t, \tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1} \Vert v_{1} \Vert _{H^{s-1}\cap L^{m}}, \\ &\bigl\Vert \vert D \vert ^{s} v(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(\tau )^{-1} \bigl(1+B(t, \tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{s}{2}} \Vert v_{1} \Vert _{H^{s-1}\cap L^{m}}, \\ &\bigl\Vert \vert D \vert ^{s-1} v_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b(\tau )^{-1}b(t)^{-1} \bigl(1+B(t,\tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{s-1}{2}-1} \\ &\phantom{\bigl\Vert \vert D \vert ^{s-1} v_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim}{} \times \Vert v_{1} \Vert _{H^{s-1}\cap L^{m}}. \end{aligned}$$

(20)

The proof of this theorem follows from [5] and [19].

3.2 Proofs

We define the norm of the solution space $X(t)$ by

$$ \bigl\Vert (u,v) \bigr\Vert _{X(t)}=\sup_{\tau \in [0,t]} \bigl\{ M_{1}(\tau,u)+M_{2}( \tau,v) \bigr\} , $$

where we will choose $M_{1}(\tau,u)$ and $M_{2}(\tau,v)$ with respect to the goals of each theorem.

Let N be the mapping on $X(t)$ defined by

$$ N: (u,v) \in X(t) \to N(u,v)= \bigl(u^{ln}+u^{nl},v^{ln}+v^{nl} \bigr), $$

where

$$\begin{aligned} &u^{ln}(t,x):= E_{1,0}(t,0,x)\ast _{(x)}u_{0}(x)+E_{1,1}(t,0,x) \ast _{(x)}u_{1}(x), \\ &u^{nl}(t,x):= \int _{0}^{t}E_{1,1}(t,\tau,x)\ast _{(x)}f(\tau,v)\,d \tau, \\ &v^{ln}(t,x):= E_{2,0}(t,0,x)\ast _{(x)}v_{0}(x)+E_{2,1}(t,0,x) \ast _{(x)}v_{1}(x), \\ &v^{nl}(t,x):= \int _{0}^{t}E_{2,1}(t,\tau,x)\ast _{(x)}g(\tau,u)\,d \tau. \end{aligned}$$

We denote by $E_{1,0}=E_{1,0}(t,0,x)$ and $E_{1,1}=E_{1,1}(t,0,x)$ the fundamental solutions to the Cauchy problem

$$ u_{tt}-\Delta u+b_{1}(t)u_{t}=0,\qquad u(0,x)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x), $$

and by $E_{2,0}=E_{2,0}(t,0,x)$ and $E_{2,1}=E_{2,1}(t,0,x)$ the fundamental solutions to the the Cauchy problem

$$ v_{tt}-\Delta v+b_{2}(t)v_{t}=0,\qquad v(0,x)=v_{0}(x),\qquad v_{t}(0,x)=v_{1}(x). $$

Our aim is to prove the estimates

$$\begin{aligned} \begin{aligned}& \bigl\Vert N(u,v) \bigr\Vert _{X(t)} \\ &\quad\lesssim \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned} \begin{aligned}& \bigl\Vert N(u,v)-N(\tilde{u}, \tilde{v}) \bigr\Vert _{X(t)}\lesssim \bigl\Vert (u,v) - (\tilde{u}, \tilde{v}) \bigr\Vert _{X(t)} \\ &\phantom{\bigl\Vert N(u,v)-N(\tilde{u}, \tilde{v}) \bigr\Vert _{X(t)}\lesssim}{}\times \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p-1} + \bigl\Vert (\tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{p-1} + \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q-1}+ \bigl\Vert ( \tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{q-1} \bigr). \end{aligned} \end{aligned}$$

(22)

We can immediately obtain from the introduced norm of the solution space $X(t)$, which will be fixed for each case, the following inequality:

$$ \bigl\Vert \bigl(u^{ln},v^{ln}\bigr) \bigr\Vert _{X(t)}\lesssim \bigl\Vert (u_{0},u_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{1}}}+ \bigl\Vert (v_{0},v_{1}) \bigr\Vert _{\mathcal{A}_{m,s_{2}}}. $$

We complete the proof of all results separately by showing (22) with the inequality

$$ \bigl\Vert \bigl(u^{nl},v^{nl}\bigr) \bigr\Vert _{X(t)}\lesssim \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, $$

(23)

which leads to (21).

Proof of Theorem 2.1

We choose the space of energy solutions

$$ X(t)= \bigl( \mathcal{C}\bigl([0,t], H^{1}\bigr)\cap \mathcal{C}^{1}\bigl([0,t], L^{2}\bigr) \bigr) ^{2} $$

with the following norms for $\tau \in (0,t]$:

$$\begin{aligned} M_{1}(\tau,u) ={}& \bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} ) } \bigl\Vert u(\tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{} +\bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+\frac{1}{2}} \bigl\Vert \nabla u( \tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{}+b_{1}(\tau ) \bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )+1} \bigl\Vert u_{t}(\tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}, \\ M_{2}(\tau,v)={}& \bigl(1+B_{2}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} ) } \bigl\Vert v(\tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{} +\bigl(1+B_{2}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+\frac{1}{2}} \bigl\Vert \nabla v( \tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{}+b_{2}(\tau ) \bigl(1+B_{2}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )+1} \bigl\Vert v_{t}(\tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}. \end{aligned}$$

To prove (23), we need to estimate all terms appearing in $\|(u^{nl},v^{nl})\|_{X(t)}$. Let us begin to estimate $\| u^{nl}_{t}(t,\cdot ) \| _{L^{2}}$. Using (19) with $m=2$ for $\tau \in [\frac{t}{2},t]$, we get

$$\begin{aligned} \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} } \lesssim{} & { \int _{0}^{\frac{t}{2}}}b_{1}(t)^{-1}b_{1}( \tau )^{-1}\bigl(1+B_{1}(t, \tau )\bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-1} \bigl\Vert f( \tau,v) \bigr\Vert _{L^{m} \cap L^{2} }\,d\tau \\ &{}+ { \int _{\frac{t}{2}}^{t}}b_{1}(t)^{-1} b_{1}(\tau )^{-1}\bigl(1+B_{1}(t, \tau ) \bigr)^{-1} \bigl\Vert f(\tau,v) \bigr\Vert _{L^{2} }\,d\tau. \end{aligned}$$

(24)

By a fractional version of the Gagliardo–Nirenberg inequality (see Proposition 4.1) and (5) we obtain

$$\begin{aligned} & \bigl\Vert f(\tau,v) \bigr\Vert _{L^{2}} \lesssim \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}( \tau,0)\bigr)^{-\frac{n}{2m}p+\frac{n}{4}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(25)

$$\begin{aligned} & \bigl\Vert f(\tau,v) \bigr\Vert _{L^{m}} \lesssim \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}}\bigl(1+B_{2}( \tau,0)\bigr)^{-\frac{n}{2m}p+\frac{n}{2m}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(26)

where we use condition (11). Plugging the last estimates into (24) and using (4), (17), and (18), we get

$$\begin{aligned} \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim{}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} { \int _{0}^{\frac{t}{2}}}b_{1}(t)^{-1}b_{1}( \tau )^{-1}\bigl(1+B_{1}(t, \tau )\bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-1} \\ &{} \times \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}( \tau,0)\bigr)^{-\frac{n}{2m}p+\frac{n}{2m}} \,d\tau \\ &{} + \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} { \int _{ \frac{t}{2}}^{t}}b_{1}(t)^{-1} b_{1}(\tau )^{-1}\bigl(1+B_{1}(t,\tau ) \bigr)^{-1} \\ &{} \times \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}( \tau,0)\bigr)^{-\frac{n}{2m}p+\frac{n}{4}} \,d\tau \\ \lesssim{}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} { \int _{0}^{\frac{t}{2}}}b_{1}(t)^{-1}b_{1}( \tau )^{-1}\bigl(1+B_{1}(t,\tau )\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1} \\ &{} \times \bigl(1+B_{1}(\tau,0)\bigr)^{(- \frac{n}{2m}p+\frac{n}{2m})\beta +\gamma _{1}} \,d\tau \\ &{} + \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} { \int _{ \frac{t}{2}}^{t}}b_{1}(t)^{-1} b_{1}(\tau )^{-1}\bigl(1+B_{1}(t,\tau ) \bigr)^{-1} \\ &{} \times \bigl(1+B_{1}(\tau,0)\bigr)^{(- \frac{n}{2m}p+\frac{n}{4})\beta +\gamma _{1}} \,d\tau \\ \lesssim{}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-1} \\ &{} \times { \int _{0}^{ \frac{t}{2}}}b_{1}(\tau )^{-1} \bigl(1+B_{1}(\tau,0)\bigr)^{(-\frac{n}{2m}p+ \frac{n}{2m})\beta +\gamma _{1}} \,d\tau \\ &{} + \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{(- \frac{n}{2m}p+\frac{n}{4})\beta +\gamma _{1}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t,\tau )\bigr)^{-1} \,d\tau. \end{aligned}$$

The last integral can be obtained from the definition of $B_{1}(t,\tau )$; indeed,

$$ \int _{\frac{t}{2}}^{t} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t, \tau )\bigr)^{-1} \,d\tau \lesssim \log \bigl(1+B_{1}(t,0)\bigr)\approx \bigl(1+B_{1}(t, \tau ) \bigr)^{\nu }, $$

where ν sufficiently small.

We distinguish two cases with respect to the value of β. If $\beta \geq 1$, then we get

$$\begin{aligned} \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} } \lesssim {}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1} \\ &{} \times { \int _{0}^{ \frac{t}{2}}}b_{1}(\tau )^{-1} \bigl(1+B_{1}(\tau,0)\bigr)^{-\frac{n}{2m}( \tilde{p}-1)+\gamma _{1}} \,d\tau \\ &{}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2m}( \tilde{p}-1)-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) \beta +\gamma _{1}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t,\tau )\bigr)^{-1} \,d\tau. \end{aligned}$$

We can conclude from $-\frac{n}{2m}(\tilde{p}-1)+\gamma _{1}<-1$, which is equivalent to $\tilde{p}>\frac{2m(\gamma _{1} +1)}{n}+1$, that ${\int _{0}^{\frac{t}{2}}}b_{1}(\tau )^{-1} (1+B_{1}( \tau,0))^{-\frac{n}{2m}(\tilde{p}-1)+\gamma _{1}} \,d\tau$ is bounded. Hence

$$ \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} }\lesssim \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1}. $$

If $0<\beta <1$, then we get

$$\begin{aligned} \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} } \lesssim{} & \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1} \\ &{} \times { \int _{0}^{ \frac{t}{2}}}b_{1}(\tau )^{-1} \bigl(1+B_{1}(\tau,0)\bigr)^{-\frac{n}{2m}( \tilde{p}-1)-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )(1- \beta )+\gamma _{1}} \,d\tau \\ &{}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2m} \tilde{p}+\frac{n}{4}+\gamma _{1}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t,\tau )\bigr)^{-1} \,d\tau \\ \lesssim{} & \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-1} \end{aligned}$$

for $\tilde{p}>\frac{2m(\gamma _{1} +1)}{n}+1$. Finally, we obtain

$$ \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} }\lesssim \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1}. $$

(27)

Analogously, we can prove that

$$\begin{aligned} &\bigl\Vert \nabla u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{1}{2} } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(28)

$$\begin{aligned} &\bigl\Vert u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}. \end{aligned}$$

(29)

For the second component $v^{nl}$, using the Gagliardo–Nirenberg inequality, from Proposition 4.1 we get for $\tau \in (0,t]$ the following estimates:

$$\begin{aligned} & \bigl\Vert g(\tau,u) \bigr\Vert _{L^{2}} \lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}} \bigl(1+B_{1}( \tau,0)\bigr)^{-\frac{n}{2m}q+\frac{n}{4}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \\ & \bigl\Vert g(\tau,u) \bigr\Vert _{L^{m}} \lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}}\bigl(1+B_{1}( \tau,0)\bigr)^{-\frac{n}{2m}q+\frac{n}{2m}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}. \end{aligned}$$

Taking into account the last estimates, we can prove, similarly to (27)–(29), the estimates

$$\begin{aligned} & \bigl\Vert v^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b_{2}(t)^{-1} \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned}$$

(30)

$$\begin{aligned} & \bigl\Vert \nabla v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{1}{2}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned}$$

(31)

$$\begin{aligned} & \bigl\Vert v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned}$$

(32)

for $\tilde{q}>\frac{2m(\gamma _{2} +1)}{n}+1$. Finally, (27)–(32) imply (23).

The proof of (22) is completely analogous to that of (21). In this way, we complete the proof of Theorem 2.1. □

Proof of Theorem 2.5

We choose the same space of energy solutions $X(t)$ with the norm $M_{2}(\tau,v)$ used in the proof of Theorem 2.5. We modify the norm $M_{1}(\tau,v)$ as follows:

$$\begin{aligned} M_{1}(\tau,u) ={}& \bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-\kappa (\tilde{p}) } \bigl\Vert u(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+\bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+\frac{1}{2}-\kappa (\tilde{p})} \bigl\Vert \nabla u( \tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+b_{1}(\tau ) \bigl(1+B_{1}(\tau,0)\bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )+1-\kappa (\tilde{p})} \bigl\Vert u_{t}(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})}, \end{aligned}$$

where $\kappa (\tilde{p}) =\gamma _{1}-\frac{n}{2m}(\tilde{p}-1)+1$. We begin the proof of (23) by estimating the norm $\| u^{nl}_{t}(t,\cdot ) \| _{L^{2}}$. Using (19) with $m=2$ for $\tau \in [\frac{t}{2},t]$ together with the Gagliardo–Nirenberg inequality and following the same steps of the proof of (27), we get

$$\begin{aligned} \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} } \lesssim {}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1} \\ &{} \times { \int _{0}^{ \frac{t}{2}}}b_{1}(\tau )^{-1} \bigl(1+B_{1}(\tau,0)\bigr)^{(-\frac{n}{2m}p+ \frac{n}{2m})\beta +\gamma _{1}} \,d\tau \\ &{}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{(-\frac{n}{2m}p+ \frac{n}{4})\beta +\gamma _{1}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t,\tau )\bigr)^{-1} \,d\tau \\ \lesssim {}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-1+\kappa ( \tilde{p})} \end{aligned}$$

for $\beta >0$. Then we have

$$ \bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} }\lesssim \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1} \bigl(1+B_{1}(t,\tau )\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1+\kappa (\tilde{p})}. $$

(33)

In the same way, we can prove

$$\begin{aligned} &\bigl\Vert \nabla u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{1}{2} + \kappa (\tilde{p})} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(34)

$$\begin{aligned} &\bigl\Vert u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) +\kappa (\tilde{p})} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}. \end{aligned}$$

(35)

Now for $v^{nl}$, using the Gagliardo–Nirenberg inequality and the definition of the solution space $X(t)$, we can prove the following estimates:

$$\begin{aligned} & \bigl\Vert g(\tau,u) \bigr\Vert _{L^{2}} \lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}} \bigl(1+B_{1}( \tau,0)\bigr)^{-\frac{n}{2m}q+\frac{n}{4}+\kappa (\tilde{p})q} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \\ & \bigl\Vert g(\tau,u) \bigr\Vert _{L^{m}} \lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}}\bigl(1+B_{1}( \tau,0)\bigr)^{-\frac{n}{2m}q+\frac{n}{2m}+\kappa (\tilde{p})q} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}. \end{aligned}$$

Taking into account the last estimates together with (19), we obtain

$$\begin{aligned} \bigl\Vert v^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2} } \lesssim {}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q} b_{2}(t)^{-1}\bigl(1+B_{2}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-1} \\ &{} \times { \int _{0}^{ \frac{t}{2}}}b_{2}(\tau )^{-1} \bigl(1+B_{2}(\tau,0)\bigr)^{(-\frac{n}{2m}q+ \frac{n}{2m}+\kappa (\tilde{p})q)\alpha +\gamma _{2}} \,d\tau \\ &{}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}b_{2}(t)^{-1} \bigl(1+B_{2}(t,0)\bigr)^{(-\frac{n}{2m}p+ \frac{n}{4}+\kappa (\tilde{p})q)\alpha +\gamma _{2}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{2}(\tau )^{-1} \bigl(1+B_{2}(t,\tau )\bigr)^{-1} \,d\tau \\ \lesssim {}& b_{2}(t)^{-1} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned}$$

where we use the condition

$$ \gamma _{2}-\frac{n}{2m}(\tilde{q}-1) +\kappa (\tilde{p})q\alpha + \varepsilon < -1, $$

which is equivalent to condition (13). Then

$$ \bigl\Vert v^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim b_{2}(t)^{-1} \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}. $$

(36)

Analogously, we can prove

$$\begin{aligned} & \bigl\Vert \nabla v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -\frac{1}{2}} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}, \end{aligned}$$

(37)

$$\begin{aligned} & \bigl\Vert v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}}\lesssim \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q}. \end{aligned}$$

(38)

Consequently, (33)–(38) imply (23).

To prove (22), we suppose the existence of $(u,v)$ and $(\tilde{u},\tilde{v})$ belonging to the space of solution $X(t)$. Then we have

$$\begin{aligned} N(u,v)-N(\tilde{u},\tilde{v}) ={}&\bigl(u^{nl}(t,x)- \tilde{u}^{nl}(t,x),v^{nl}(t,x)- \tilde{v}^{nl}(t,x) \bigr) \\ ={}& \biggl( \int _{0}^{t}E_{1}(t,\tau,x)\ast _{(x)} \bigl(f(\tau,v)- f(\tau,\tilde{v}) \bigr)\,d\tau, \\ &{} \int _{0}^{t}E_{1}(t,\tau,x)\ast _{(x)} \bigl(g(\tau,u)- g(\tau,\tilde{u}) \bigr)\,d\tau \biggr). \end{aligned}$$

Similarly to (25) and (26), using (5) and (6), we can prove the following estimates:

$$\begin{aligned} \bigl\Vert f(\tau,v)- f(\tau,\tilde{v}) \bigr\Vert _{L^{2}}\lesssim {}&\bigl(1+B_{1}( \tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}(\tau,0)\bigr)^{-\frac{n}{2m}p+ \frac{n}{4}} \end{aligned}$$

(39)

$$\begin{aligned} &{} \times \Vert v-\widetilde{v} \Vert _{X(t)} \bigl( \Vert v \Vert _{X(t)}^{p-1} + \Vert \widetilde{v} \Vert _{X(t)}^{p-1} \bigr), \\ \bigl\Vert f(\tau,v)- f(\tau,\tilde{v}) \bigr\Vert _{L^{m}}\lesssim {}&\bigl(1+B_{1}( \tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}(\tau,0)\bigr)^{-\frac{n}{2m}p+ \frac{n}{2m}} \end{aligned}$$

(40)

$$\begin{aligned} &{} \times \Vert v-\widetilde{v} \Vert _{X(t)} \bigl( \Vert v \Vert _{X(t)}^{p-1} + \Vert \widetilde{v} \Vert _{X(t)}^{p-1} \bigr), \\ \bigl\Vert g(\tau,u)- f(\tau,\tilde{u}) \bigr\Vert _{L^{2}}\lesssim {}&\bigl(1+B_{2}( \tau,0)\bigr)^{\gamma _{2}} \bigl(1+B_{1}(\tau,0)\bigr)^{-\frac{n}{2m}q+ \frac{n}{4}+\kappa (\tilde{p})q } \end{aligned}$$

(41)

$$\begin{aligned} &{} \times \Vert u-\widetilde{u} \Vert _{X(t)} \bigl( \Vert u \Vert _{X(t)}^{q-1} + \Vert \widetilde{u} \Vert _{X(t)}^{q-1} \bigr), \\ \bigl\Vert g(\tau,u)- f(\tau,\tilde{u}) \bigr\Vert _{L^{m}}\lesssim{} &\bigl(1+B_{2}( \tau,0)\bigr)^{\gamma _{2}} \bigl(1+B_{1}(\tau,0)\bigr)^{-\frac{n}{2m}q+ \frac{n}{2m} +\kappa (\tilde{p})q} \\ &{} \times \Vert u-\widetilde{u} \Vert _{X(t)} \bigl( \Vert u \Vert _{X(t)}^{q-1} + \Vert \widetilde{u} \Vert _{X(t)}^{q-1} \bigr). \end{aligned}$$

(42)

Analogously to (33)–(38), using (39)–(42), we can get

$$\begin{aligned} & \biggl\Vert \nabla ^{j}\partial _{t}^{l} \int _{0}^{t}E_{1}(t,\tau,x)\ast _{(x)} \bigl(f(\tau,v)- f(\tau, \widetilde{v}) \bigr) \,d\tau \biggr\Vert _{L^{2}} \\ &\quad \lesssim b(t)^{-l}\bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{j}{2}-l} \\ &\qquad{}\times \sup_{\tau \in [0,t]} M_{2}(\tau,v- \widetilde{v}) \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p-1}+ \bigl\Vert (\tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{p-1} \bigr), \end{aligned}$$

(43)

$$\begin{aligned} &\biggl\Vert \nabla ^{j}\partial _{t}^{l} \int _{0}^{t}E_{1}(t,\tau,x)\ast _{(x)} \bigl(g(\tau,u)- g(\tau, \widetilde{u}) \bigr) \,d\tau \biggr\Vert _{L^{2}} \\ &\quad\lesssim b(t)^{-l}\bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}} \bigl(1+B^{1}(t,0) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{j}{2}-l+\kappa (\tilde{p})q } \\ &\qquad{}\times \sup_{\tau \in [0,t]}M_{1}(\tau,u- \widetilde{u}) \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p-1}+ \bigl\Vert (\tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{p-1} \bigr), \end{aligned}$$

(44)

where $j+l\leq 1$. The proof is completed. □

Proof of Theorem 2.7

Let us choose the space of energy solutions with suitable regularity

$$ X(t)= \bigl(\mathcal{C}\bigl([0,t], H^{s_{1}}\bigr)\cap \mathcal{C}^{1}\bigl([0,t], H^{s_{1}-1}\bigr) \bigr)\times \bigl( \mathcal{C}\bigl([0,t], H^{s_{2}}\bigr)\cap \mathcal{C}^{1} \bigl([0,t], H^{s_{2}-1}\bigr) \bigr) $$

with the norm

$$ \bigl\Vert (u,v) \bigr\Vert _{X(t)}=\sup_{\tau \in [0,t]} \bigl\{ M_{1}(\tau,u)+M_{2}( \tau,v) \bigr\} , $$

where

$$\begin{aligned} M_{1}(\tau,u) ={} & \bigl(1+B_{1} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert u(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+b_{1}(\tau ) \bigl(1+B_{1} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+1} \bigl\Vert u_{t}(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+b_{1}(\tau ) \bigl(1+B_{1} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+\frac{s_{1}-1}{2}+1} \bigl\Vert \vert D \vert ^{s_{1}-1} u_{t}( \tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{}+ \bigl(1+B_{1} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )+\frac{s_{1}}{2}} \bigl\Vert \vert D \vert ^{s_{1}} u(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \end{aligned}$$

and

$$\begin{aligned} M_{2}(\tau,v) = {}& \bigl(1+B_{2} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) } \bigl\Vert v(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+b_{2}(\tau ) \bigl(1+B_{2} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+1} \bigl\Vert v_{t}(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})} \\ &{}+b_{2}(\tau ) \bigl(1+B_{2} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )+\frac{s_{2}-1}{2}+1} \bigl\Vert \vert D \vert ^{s_{2}-1} v_{t}( \tau,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &{} + \bigl(1+B_{2} (\tau,0) \bigr)^{\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )+\frac{s_{2}}{2}} \bigl\Vert \vert D \vert ^{s_{2}} v(\tau,\cdot ) \bigr\Vert _{L^{2}( \mathbb{R}^{n})}. \end{aligned}$$

To prove (23), we show how to estimate the norms $\||D|^{s_{1}-1}u^{nl}_{t}(t,\cdot )\|_{L^{2}(\mathbb{R}^{n})}$ and $\||D|^{s_{2}-1}\times v^{nl}_{t}(t,\cdot )\|_{L^{2}(\mathbb{R}^{n})}$. From estimate (20) it follows that

$$\begin{aligned} & \bigl\Vert \vert D \vert ^{s_{1}-1}u^{nl}_{t}(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \\ &\quad \lesssim { \int _{0}^{\frac{t}{2}}}b_{1}(\tau )^{-1}b_{1}(t)^{-1}\bigl(1+B_{1}(t, \tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{s_{1}-1}{2}-1} \\ &\qquad{} \times \bigl\Vert f(\tau,v) \bigr\Vert _{L^{m}(\mathbb{R}^{n})\cap L^{2}( \mathbb{R}^{n})\cap \dot{H}^{s_{1}-1} (\mathbb{R}^{n})}\,d\tau \\ & \qquad{}+ { \int _{\frac{t}{2}}^{t}}b_{1}(\tau )^{-1}b_{1}(t)^{-1}\bigl(1+B_{1}(t, \tau ) \bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) - \frac{s_{1}-1}{2}-1} \\ &\qquad{} \times \bigl\Vert f(\tau,v) \bigr\Vert _{L^{m}(\mathbb{R}^{n})\cap L^{2}( \mathbb{R}^{n})\cap \dot{H}^{s_{1}-1} (\mathbb{R}^{n})}\,d\tau. \end{aligned}$$

Under the assumptions of Theorem 2.7 and the choice of the above introduced norm, for $0\leq \tau \leq t$, the inequalities (25) and (26) remain true. We calculate the norm

$$ \bigl\Vert f(\tau,v) \bigr\Vert _{\dot{H}^{s_{1}-1}}. $$

Using (56) and (57), for $p > \lceil s_{1}-1\rceil$ and $0\leq \tau \leq t$, we get the following estimate:

$$\begin{aligned} \bigl\Vert f(\tau,v) \bigr\Vert _{\dot{H}^{s_{1}-1}} \lesssim {}& \bigl(1+B_{1}(\tau,0)\bigr)^{ \gamma _{1}} \bigl\Vert v(\tau,\cdot ) \bigr\Vert ^{p-1} _{L^{q_{1}}} \bigl\Vert \vert D \vert ^{s_{1}-1} (\tau,\cdot ) \bigr\Vert _{L^{q_{2}}} \\ \lesssim {}&\bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl\Vert v( \tau,\cdot ) \bigr\Vert ^{(p-1)(1-\theta _{1})} _{L^{2}} \\ &{} \times \bigl\Vert \vert D \vert ^{s_{2}}v(\tau,\cdot ) \bigr\Vert ^{(p-1)\theta _{1}}_{L^{2}} \bigl\Vert v(\tau,\cdot ) \bigr\Vert ^{1-\theta _{2}}_{L^{2}} \bigl\Vert \vert D \vert ^{s_{2}}v( \tau,\cdot ) \bigr\Vert ^{\theta _{2}} _{L^{2}} \\ \lesssim {}& \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}(\tau,0)\bigr)^{- \frac{n}{2m}p+\frac{n}{4}-\frac{s_{1}-1}{2}} \bigl\Vert (u,v) \bigr\Vert ^{p}_{X(t)}, \end{aligned}$$

where

$$\begin{aligned} &\frac{p-1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{2},\qquad \theta _{1}= \frac{n}{s} \biggl( \frac{1}{2}-\frac{1}{q_{1}} \biggr) \in [0,1],\\ & \theta _{2}=\frac{n}{s_{2}} \biggl( \frac{1}{2}- \frac{1}{q_{2}} \biggr) + \frac{s_{1}-1}{s_{2}} \in \biggl[ \frac{s_{1}-1}{s_{2}},1 \biggr]. \end{aligned}$$

To satisfy the last conditions for the parameters $\theta _{1}$ and $\theta _{2}$, we choose $q_{2}=\frac{2n}{n-2}$ and $q_{1}=n(p-1)$. This choice implies the condition

$$ 1+\frac{2}{n} \leq p\leq 1+\frac{2}{n-2s_{2}}. $$

Consequently, for $\tau \in (0,t]$, we obtain the estimate

$$ \bigl\Vert f(\tau,v)|_{\dot{H}^{s_{1}-1}}\lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{- \frac{n}{2m}p+\frac{n}{4}-\frac{s_{2}-1}{2}} \bigr\Vert (u,v) \|^{p}_{X(t)}. $$

(45)

Summarizing all estimates implies

$$\begin{aligned} \bigl\Vert \vert D \vert ^{s_{1}-1}u^{nl}_{t}(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim{}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{s_{1}-1}{2}-1} \\ &{} \times { \int _{0}^{\frac{t}{2}}}b_{1}(\tau )^{-1} \bigl(1+B_{1}(\tau,0)\bigr)^{(-\frac{n}{2m}p+\frac{n}{2m})\beta +\gamma _{1}} \,d \tau \\ &{}+ \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{(-\frac{n}{2m}p+ \frac{n}{4})\beta +\gamma _{1}} \\ &{} \times { \int _{\frac{t}{2}}^{t}} b_{1}(\tau )^{-1} \bigl(1+B_{1}(t, \tau )\bigr)^{-\frac{s_{1}-1}{2}-1} \,d\tau \\ \lesssim{}& \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p} b_{1}(t)^{-1}\bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{s_{1}-1}{2}-1 }, \end{aligned}$$

where $\tilde{p}>\frac{2m}{n} (\frac{s_{1}+1+2\gamma _{1}}{2} ) +1$.

Then

$$ \bigl\Vert \vert D \vert ^{s_{1}-1}u^{nl}_{t}(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})} \lesssim b_{1}(t)^{-1} \bigl(1+B_{1}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}- \frac{1}{2} )-\frac{s_{1}-1}{2}-1 } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}. $$

(46)

Under the first condition of (14), in the same way, we can prove the following estimates:

$$\begin{aligned} &\bigl\Vert u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{1}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(47)

$$\begin{aligned} &\bigl\Vert u^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim b_{1}(t)^{-1}\bigl(1+B_{1}(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1 } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(48)

$$\begin{aligned} &\bigl\Vert \vert D \vert ^{s_{1}}u^{nl}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{1}(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{s_{1}}{2} } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}. \end{aligned}$$

(49)

Using the second condition of (14), we get

$$\begin{aligned} &\bigl\Vert \vert D \vert ^{s_{2}-1} v^{nl}_{t}(t, \cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim b_{2}(t)^{-1} \bigl(1+B_{2}(t,0)\bigr)^{-\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{s_{2}-1}{2}-1 } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(50)

$$\begin{aligned} &\bigl\Vert v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{2}(t,0)\bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )} \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(51)

$$\begin{aligned} &\bigl\Vert v^{nl}_{t}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim b_{2}(t)^{-1}\bigl(1+B_{2}(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} ) -1 } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}, \end{aligned}$$

(52)

$$\begin{aligned} &\bigl\Vert \vert D \vert ^{s_{2}} v^{nl}(t,\cdot ) \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{2}(t,0) \bigr)^{- \frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )-\frac{s_{2}}{2} } \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p}. \end{aligned}$$

(53)

From (46)–(53) we get (23), which completes the proof of (21).

To prove (22), we use the same steps used in the previous proof. Indeed, from the fractional Leibniz rule (see Proposition 4.2) and the fractional chain rule (see Proposition 4.3) we may conclude for $0\leq \tau \leq t$ the following estimates:

$$\begin{aligned} \bigl\Vert f(\tau,v)- f(\tau,\tilde{v}) \bigr\Vert _{\dot{H}^{s_{1}-1} }\lesssim{} & \bigl(1+B_{1}( \tau,0)\bigr)^{\gamma _{1}} \bigl(1+B_{2}(\tau,0)\bigr)^{ -\frac{n}{2m}p+ \frac{n}{4}-\frac{s_{2}-1}{2}} \\ &{} \times \sup_{\tau \in [0,t]} M_{2}(\tau,v- \widetilde{v}) \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p-1}+ \bigl\Vert (\tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{p-1} \bigr), \end{aligned}$$

(54)

and

$$\begin{aligned} \bigl\Vert g(\tau,u)- g(\tau,\tilde{u}) \bigr\Vert _{\dot{H}^{s_{2}-1} }\lesssim {}& \bigl(1+B_{2}( \tau,0)\bigr)^{\gamma _{2}} \bigl(1+B_{1}(\tau,0)\bigr)^{-\frac{n}{2m}q+\frac{n}{4}- \frac{s_{1}-1}{2}} \\ &{} \times \sup_{\tau \in [0,t]} M_{1}(\tau,u- \widetilde{u}) \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q-1}+ \bigl\Vert (\tilde{u},\tilde{v}) \bigr\Vert _{X(t)}^{q-1} \bigr), \end{aligned}$$

(55)

where we use condition (15). From (39)–(42) without loss of decay and (54)–(55) we can complete the proof. □

Remark 3.4

Theorem 2.8 can be proved by using a similar approach as in the proof of Theorem 2.7, with modifications in the estimates of some terms. Then using Proposition 4.4, Corollary 4.5, and Lemma 4.6, we can obtain the estimates

$$\begin{aligned} &\bigl\Vert f(\tau,v) \bigr\Vert _{\dot{H}^{s_{1}-1}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{1}( \tau,0)\bigr)^{\gamma _{1}}\bigl(1+B_{2}( \tau,0)\bigr)^{ -\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )p-\frac{s_{1}-1}{2}-\frac{s^{*}}{2}(p-1)} \bigl\Vert (u,v) \bigr\Vert ^{p}_{X(t)}, \\ &\bigl\Vert g(\tau,u) \bigr\Vert _{\dot{H}^{\tilde{s}_{2}-1}(\mathbb{R}^{n})}\lesssim \bigl(1+B_{2}( \tau,0)\bigr)^{\gamma _{2}}\bigl(1+B_{1}( \tau,0)\bigr)^{ -\frac{n}{2} ( \frac{1}{m}-\frac{1}{2} )p-\frac{\tilde{s}_{2}-1}{2}- \frac{s^{*}}{2}(q-1)} \bigl\Vert (u,v) \bigr\Vert ^{q}_{X(t)}, \\ & \bigl\Vert f(\tau,v)- f(\tau,\tilde{v}) \bigr\Vert _{\dot{H}^{s_{1}-1}( \mathbb{R}^{n}) }\\ &\quad \lesssim \bigl(1+B_{2}(\tau,0)\bigr)^{ -\frac{n}{2m}p+ \frac{n}{4}(p-1)-\frac{s_{1}-1}{2}-\frac{s^{*}}{2}(p-1)} \\ &\qquad{} \times \bigl(1+B_{1}(\tau,0)\bigr)^{\gamma _{1}} \bigl\Vert (u,v) - ( \tilde{u},\tilde{v}) \bigr\Vert _{X(t)} \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{p-1}+ \bigl\Vert ( \tilde{u}, \tilde{v}) \bigr\Vert _{X(t)}^{p-1} \bigr), \\ & \bigl\Vert g(\tau,u)- g(\tau,\tilde{u}) \bigr\Vert _{\dot{H}^{\tilde{s}_{2}-1}( \mathbb{R}^{n}) } \\ &\quad\lesssim \bigl(1+B_{1}(\tau,0)\bigr)^{ -\frac{n}{2m}p+ \frac{n}{4}(q-1)-\frac{\tilde{s}_{2}-1}{2}-\frac{s^{*}}{2}(q-1)} \\ &\qquad{} \times \bigl(1+B_{2}(\tau,0)\bigr)^{\gamma _{2}} \bigl\Vert (u,v) - ( \tilde{u},\tilde{v}) \bigr\Vert _{X(t)} \bigl( \bigl\Vert (u,v) \bigr\Vert _{X(t)}^{q-1}+ \bigl\Vert ( \tilde{u}, \tilde{v}) \bigr\Vert _{X(t)}^{q-1} \bigr). \end{aligned}$$

Using these estimates, provided that condition (16) is satisfied, we can follow steps in the proof of Theorem 2.7 to complete our proof.

Appendix

Here we state some inequalities, which come into play in our proofs.

Proposition 4.1

Let $1< p,p_{0},p_{1}<\infty$, $\sigma >0$, and $s\in [0,\sigma )$. Then the following fractional Gagliardo–Nirenberg inequality holds for all $u\in L^{p_{0}} \cap \dot{H} _{p_{1}}^{\sigma }$:

$$ \Vert u \Vert _{\dot{H} _{p}^{s}}\lesssim \Vert u \Vert _{L^{p_{0}}}^{1-\theta } \Vert u \Vert _{ \dot{H} _{p_{1}}^{\sigma }}^{\theta }, $$

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where

$$ \theta =\theta _{s,\sigma }:= \frac{\frac{1}{p_{0}}-\frac{1}{p}+\frac{s}{n}}{\frac{1}{p_{0}}-\frac{1}{p_{1}}+\frac{\sigma }{n}} \quad\textit{and}\quad \frac{s}{\sigma }\leq \theta \leq 1. $$

For the proof, see [11] and [2, 8–10, 14, 15].

Proposition 4.2

Let $s>0$, $1\leq r\leq \infty$, and $1< p_{1},p_{2},q_{1},q_{2}\leq \infty$ satisfy the relation

$$ \frac{1}{r}= \frac{1}{p_{1}}+\frac{1}{p_{2}}= \frac{1}{q_{1}}+ \frac{1}{q_{2}}. $$

Then we have the following fractional Leibniz rule:

$$ \bigl\Vert \vert D \vert ^{s}(fg) \bigr\Vert _{L^{r}} \lesssim \bigl\Vert \vert D \vert ^{s}f \bigr\Vert _{L^{p_{1}}} \Vert g \Vert _{L^{p_{2}}}+ \Vert f \Vert _{L^{q_{1}}} \bigl\Vert \vert D \vert ^{s}g \bigr\Vert _{L^{q_{2}}} $$

for all $f\in \dot{H} _{p_{1}}^{s}\cap L^{q_{1}}$ and $g\in \dot{H} _{q_{2}}^{s}\cap L^{p_{2}}$.

For more details concerning fractional Leibniz rule, see [8].

Proposition 4.3

Let us choose $s>0,p>\lceil s \rceil$, and $1< r,r_{1},r_{2}<\infty$ satisfying

$$ \frac{1}{r}=\frac{p-1}{r_{1}}+\frac{1}{r_{2}}. $$

Let $F(u)$ be one of the functions $|u|^{p}, \pm |u|^{p-1}u$. Then we have the following fractional chain rule:

$$ \bigl\Vert \vert D \vert ^{s}F(u) \bigr\Vert _{L^{r}}\lesssim \Vert u \Vert _{L^{r_{1}}}^{p-1} \bigl\Vert \vert D \vert ^{s}u \bigr\Vert _{L^{r_{2}}}. $$

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For the proof, see [24].

Proposition 4.4

Let $p>1$ and $u\in H^{s}_{m}$, where $s\in (\frac{n}{m},p)$. Then we have the following estimates:

$$\begin{aligned} &\bigl\Vert \vert u \vert ^{p} \bigr\Vert _{H^{s}_{m}} \lesssim \Vert u \Vert _{ H^{s}_{m}} \Vert u \Vert ^{p-1}_{L^{ \infty }}, \\ &\bigl\Vert u \vert u \vert ^{p-1} \bigr\Vert _{H^{s}_{m}} \lesssim \Vert u \Vert _{ H^{s}_{m}} \Vert u \Vert ^{p-1}_{L^{ \infty }}. \end{aligned}$$

For the proof, see [26].

From Proposition 4.4 we can derive the following corollary.

Corollary 4.5

Under the assumptions of Proposition 4.4, we have

$$\begin{aligned} &\bigl\Vert \vert u \vert ^{p} \bigr\Vert _{\dot{H}^{s}_{m}} \lesssim \Vert u \Vert _{ \dot{H}^{s}_{m}} \Vert u \Vert ^{p-1}_{L^{ \infty }}, \\ &\bigl\Vert u \vert u \vert ^{p-1} \bigr\Vert _{\dot{H}^{s}_{m}} \lesssim \Vert u \Vert _{ \dot{H}^{s}_{m}} \Vert u \Vert ^{p-1}_{L^{\infty }}. \end{aligned}$$

For the proof, see [6] and [25].

Lemma 4.6

Let $0<2s^{*}<n<2s$. Then for any function $f\in \dot{H}^{s^{*}}\cap \dot{H}^{s}$, we have the estimate

$$ \Vert f \Vert _{L^{\infty }}\leq \Vert f \Vert _{\dot{H}^{s^{*}}}+ \Vert f \Vert _{\dot{H}^{s}}. $$

For the proof, see [4].