1. Introduction
The well known “Growth” phenomenon is one of the most important phenomena of asymptotic behavior, where many researches omit from its study especially when it comes from the evolution problems. It gives us very important information to know the behavior of equation when time arrives at infinity, it differs from global existence and blow up in both mathematically and in applications point of view. Although the interest of the scientific community for the study of delayed problems is fairly recent, multiple techniques have already been explored in depth.
In this direction, we are concerned with the delayed damped system
\begin{equation}
\left\{
\begin{array}{l}
u_{tt}-\Delta u-\omega\Delta u_{t}+\displaystyle\int_{0}^{t }\varpi(t-q) \Delta u(q)
dq\\
+\mu _{1}u_{t} +\displaystyle\int_{\tau _{1}}^{\tau _{2}}\vert\mu_{2} (q)\vert u_{t}(x, t-q) dq=b\vert u\vert ^{p-2}.u, \ x\in \Omega, t>0, \\
u\left( x, t\right) =0, x\in \partial \Omega, \\
u_{t}\left( x, -t \right) =f_{0}\left( x, t \right), \ (x, t)\in \Omega\times\left(
0, \tau_{2} \right), \\
u\left( x, 0\right) =u_{0}\left( x\right), u_{t}\left( x, 0\right)
=u_{1}\left( x\right), x\in \Omega,%
\end{array}%
\right. \label{system1}
\end{equation}
(1)
where \(\omega,b, \mu_{1}\) are positive constants, \(p>2\) and \(\tau_{1}, \tau_{2}\) are the time delay with \(0\leq\tau_{1}< \tau_{2}\) and \(\mu_{2}\) is bounded function and \(\varpi\) is a differentiable function.
It is well known that viscous materials are the opposite of elastic materials which have the capacity to store and dissipate mechanical energy. As the mechanical properties of these viscous substances are of great importance when they appear in many applications of other applied sciences. Many searchers have paid attention to this problem.
In the absence of the strong damping \(\Delta u_{t}\), that is for \(w=0\) and in absence of the distributed delay term. Our problem (1) has been investigated by many authors and results on the local/global existence, and stability have been established. See for example [1, 2, 3, 4]. In [5], the authors looked into the following system
\begin{equation}
u_{tt}-\Delta u+\displaystyle\int_{0}^{t }\varpi(t-s) \Delta u(s)
ds+a(x)u_{t}+\vert u\vert ^{\gamma}u=0,
\end{equation}
(2)
where decay result of an exponential rate was showed.
In [
6], Song and Xue considered with the following viscoelastic equation with strong damping:
\begin{equation}
\left\{
\begin{array}{l}
u_{tt}-\Delta u+\displaystyle\int_{0}^{\infty }g(t-s) \Delta u(s)
ds-\Delta u_{t} =\vert u\vert ^{p-2}.u, \ x\in \Omega, t>0, \\
u\left( x, 0\right) =u_{0}\left( x\right), u_{t}\left( x, 0\right)
=u_{1}\left( x\right).\label{p1.1}
\end{array}
\right.
\end{equation}
(3)
The authors showed, under suitable conditions on \(g\), that there
were solutions of (3) with arbitrarily high initial energy that blow up in a finite time. For the same Problem (3), in [
7], Song and Zhong showed that there were solutions of (3) with positive initial energy that blew up in finite time. In [
8], Zennir considered with the following viscoelastic equation with strong damping:
\begin{equation}
\left\{
\begin{array}{l}
u_{tt}-\Delta u-\omega\Delta u_{t}+\displaystyle\int_{0}^{t }g(t-s) \Delta u(s)
ds\\
+a\vert u_{t}\vert ^{m-2}.u_{t} =\vert u\vert ^{p-2}.u, \ x\in \Omega, t>0, \\
u\left( x, 0\right) =u_{0}\left( x\right), u_{t}\left( x, 0\right)
=u_{1}\left( x\right), \ x\in \Omega\\
u(x, t)=0, \ x\in \partial\Omega.
\end{array}%\label{p1.1}
\right.
\end{equation}
(4)
They proved the exponential growth result under suitable assumptions.
In [
9] the authors considered the following problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms
\begin{equation}
\left\{
\begin{array}{l}
u_{tt}-\Delta u+\int_{0}^{\infty }g(s) \Delta u(t-s)
ds-\varepsilon_{1}\Delta u_{t}+\varepsilon_{2}u_{t}\vert u_{t}\vert ^{m-2}=\varepsilon_{3}u\vert u\vert ^{p-2}, x\in \Omega, t>0, \\
u(x, t)=0, \ x\in \partial\Omega, t>0\\
u\left( x, 0\right) =u_{0}\left( x\right), u_{t}\left( x, 0\right)
=u_{1}\left( x\right), \ x\in \Omega.%
\end{array}%
\right. \label{system8}
\end{equation}
(5)
They proved a blow up result if \(p>m\) and established the global existence.
In this article, we investigated Problem (1), in which all the damping mechanism have been considered
in the same time, these assumptions make our problem different form those studied in the literature, specially the Exponential Growth of solutions. We will prove that if the initial energy \(E(0)\) of our solutions is negative (this means that our initial data are large enough), then our local solutions in bounded and
\begin{equation}
\Vert u\Vert_{p}^{p}\rightarrow\infty,
\end{equation}
(6)
as \(t\) tends to \(\infty\), used idea in [
10,
11,
12,
13].
Our aim in the present work is to extend the existing exponential growth results to strong damping for a viscoelastic problem with distributed delay under the following assumptions:
- (A1) \(\varpi \in(\mathbb{R}_{+}, \mathbb{R}_{+})\) is decreasing function so that
\begin{equation}
\varpi(t)\geq0\hspace{0.3cm}, \ 1-\int_{0}^{\infty }\varpi\left( q\right) dq=l>0. \label{A1}
\end{equation}
(7)
- (A2) There exists a constant \(\xi>0\) such that
\begin{equation}
\varpi^{\prime }\left( t\right) \leq -\xi \varpi\left( t\right)
\hspace{0.3cm}, \ t\geq 0. \label{A2}
\end{equation}
(8)
- (A3) \(\mu _{2}:[\tau_{1}, \tau_{2}]\rightarrow\mathbb{R}\) is bounded function so that
\begin{equation}
\Big(\frac{2\delta-1}{2}\Big)\int_{\tau_{1}}^{\tau_{2}}\vert\mu _{2}(q)\vert dq\leq\mu _{1}\hspace{0.3cm}, \ \delta>\frac{1}{2}.\label{A3}
\end{equation}
(9)
2. Main results
First, as in [
14], we introduce the new variable
\begin{equation*}
y(x, \rho, q, t)=u_{t}(x, t-q\rho),
\end{equation*}
then we obtain
\begin{equation}
\left\{
\begin{array}{l}
qy_{t}(x, \rho, q, t)+y_{\rho}(x, \rho, q, t)=0\\ \\
y(x, 0, q, t)=u_{t}(x, t).
\end{array}%
\right. \label{e1.1}
\end{equation}
(10)
Let us denote by
\begin{equation}
(\varpi o u)=\int_{\Omega}\int_{0}^{t}\varpi(t-q)\vert u(t)-u(q)\vert^{2}dq. \label{e1.3}
\end{equation}
(11)
Therefore, Problem (1) takes the form
\begin{equation}
\left\{
\begin{array}{l}
u_{tt}-\Delta u-\omega\Delta u_{t}+\displaystyle\int_{0}^{t}\varpi(t-q) \Delta u(q)
dq
+\mu _{1}u_{t} +\displaystyle\int_{\tau _{1}}^{\tau _{2}}\vert\mu_{2} (q)\vert y(x, 1, q, t) dq=b\vert u\vert ^{p-2}u, \ x\in \Omega, t>0, \\
qy_{t}(x, \rho, q, t)+y_{\rho}(x, \rho, q, t)=0,
\end{array}%
\right. \label{sys1.1}
\end{equation}
(12)
%
with initial and boundary conditions
\begin{equation}
\left\{
\begin{array}{l}
u( x, t) =0, \ x\in \partial \Omega, \\
y( x, \rho, q, 0) =f_{0}\left( x, q\rho \right), \\
u\left( x, 0\right) =u_{0}\left( x\right), u_{t}\left( x, 0\right)
=u_{1}\left( x\right),
\end{array}%
\right. \label{sys1.2}
\end{equation}
(13)
%
where
$$(x, \rho, q, t)\in \Omega\times(0, 1)\times (\tau_{1}, \tau_{2})\times(0, \infty).$$
We state without proof the local existence theorem that can be established by combining arguments of [
15].
Theorem 1.
Assume (7), (8) and (9) holds. Let
\begin{equation}
\begin{cases}
2< p< \dfrac{2n-2}{n-2}, &n\geq 3;\\
p\geq 2, & n=1,2.
\end{cases}\label{a1.1}
\end{equation}
(14)
Then for any initial data
$$(u_{0},u_{1},f_{0})\in \mathcal{H}\hspace{0.2cm}/\hspace{0.2cm}\mathcal{H}= H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\times L^{2}(\Omega\times(0, 1)\times(\tau_{1}, \tau_{2})),$$
with compact support, Problem (13) has a unique solution
$$u\in C([0, T]; \mathcal{H}),$$
for some \(T>0\).
In the next theorem we give the global existence result, its proof based on the potential well depth
method in which the concept of so-called stable set appears, where we show that if we restrict our initial
data in the stable set, then our local solution obtained is global in time. One can make use of arguments in [
16].
Theorem 2.
Suppose that (7), (8), (9) and (14) hold. If \(u_{0}\in W\), \(u_{1}\in H^{1}_{0}(\Omega)\) and
\begin{equation}
\frac{bC_{*}^{p}}{l}\Big(\frac{2p}{(p-2)l}E(0)\Big)^{\frac{p-2}{2}}< 1, \label{m1.2}
\end{equation}
(15)
where \(C_{*}\) is the best Poincare’s constant. Then the local solution \(u(t, x)\) is global in time.
Lemma 1.
Assume (7), (8), (9) and (14) hold, let \(u(t)\) be a solution of
(12), then \(\mathcal{E}(t)\) is non-increasing, that is
\begin{align}
\mathcal{E}(t)&=\frac{1}{2}\Vert u_{t}\Vert_{2}^{2}+\frac{1}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}+\frac{1}{2}(\varpi o\nabla u)+\frac{1}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx-\frac{b}{p}\Vert u\Vert_{p}^{p}.\notag\\&\label{sys2.1}
\end{align}
(16)
satisfies
\begin{equation}
\mathcal{E}(t)\leq -c_{1}\Big(\Vert u_{t}\Vert_{2}^{2}+\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dqdx\Big).\label{sys2.2}
\end{equation}
(17)
Proof.
By multiplying the Equation (12)\(_{1}\) by \(u_{t}\) and integrating over \(\Omega\), we get
\begin{eqnarray}
&&\frac{d}{dt}\Big\{\frac{1}{2}\Vert u_{t}\Vert_{2}^{2}+\frac{1}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}+\frac{1}{2}(\varpi o\nabla u)-\frac{b}{p}\Vert u\Vert_{p}^{p}\Big\}\notag\\
&&=-\mu_{1}\Vert u_{t}\Vert_{2}^{2}
-\int_{\Omega}u_{t}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y(x, 1, q, t)dqdx+\frac{1}{2}(\varpi’o\nabla u)-\frac{1}{2}\varpi(t)\Vert \nabla u\Vert_{2}^{2}-\omega\Vert \nabla u_{t}\Vert_{2}^{2},\label{sys2.3}
\end{eqnarray}
(18)
and, we have
\begin{eqnarray}
&&\frac{d}{dt}\frac{1}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx=-\frac{1}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}2\vert \mu_{2}(q)\vert yy_{\rho}dqd\rho dx\notag\\
&&=\frac{1}{2}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 0, q, t)dqdx-\frac{1}{2}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dq dx\notag\\
& &=\frac{1}{2}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)dq\Big)\Vert u_{t}\Vert_{2}^{2}-\frac{1}{2}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dq dx.\label{sys2.5}
\end{eqnarray}
(19)
Then, we get
\begin{eqnarray}
&&\frac{d}{dt}\mathcal{E}(t)=-\mu_{1}\Vert u_{t}\Vert_{2}^{2}-\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert u_{t}y(x, 1, q, t)dqdx+\frac{1}{2}(\varpi’o\nabla u)\notag\\
&&-\frac{1}{2}\varpi(t)\Vert \nabla u\Vert_{2}^{2}-\omega\Vert \nabla u_{t}\Vert_{2}^{2}
+\frac{1}{2}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)dq\Big)\Vert u_{t}\Vert_{2}^{2}-\frac{1}{2}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dq dx.\label{sys2.6}
\end{eqnarray}
(20)
By (18) and (19), we get (16). Further using Young’s inequality, (7), (8) and (9) in (20), we obtain (17).
Now we are ready to state and prove our main result. For this purpose, we define
\begin{eqnarray}\label{sys2.12}
H(t)&=&-\mathcal{E}(t)\\&=&\frac{b}{p}\Vert u\Vert_{p}^{p}-\frac{1}{2}\Vert u_{t}\Vert_{2}^{2}-\frac{1}{2}(1-\int_{0}^{t}\varpi(q)dq)\Vert \nabla u\Vert_{2}^{2}-\frac{1}{2}(\varpi o\nabla u)-\frac{1}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\notag
\end{eqnarray}
(21)
Theorem 3.
Suppose that (7)-(9) and (14). Assume
further that \(E(0)< 0\) holds. Then the unique local solution of problem (12) grows
exponentially.
Proof.
From (16), we have
\begin{equation}
\mathcal{E}(t)\leq \mathcal{E}(0)\leq 0. \label{sys2.13}
\end{equation}
(22)
Hence
\begin{eqnarray}
H'(t)=-\mathcal{E}'(t)&\geq & c_{1}\Big(\Vert u_{t}\Vert_{2}^{2}+\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dqdx\Big)\notag\\
&\geq&c_{1}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dqdx\geq0, \label{sys2.14}
\end{eqnarray}
(23)
and
\begin{equation}
0\leq H(0)\leq H(t)\leq \frac{b}{p}\Vert u\Vert^{p}_{p}. \label{sys2.15}
\end{equation}
(24)
We set
\begin{equation}
\mathcal{K}(t)=H(t)+\varepsilon \int_{\Omega}uu_{t}dx+\frac{\varepsilon\mu_{1}}{2}\int_{\Omega}u^{2}dx+\frac{\varepsilon\omega}{2}\int_{\Omega}(\nabla u)^{2}dx, \label{sys2.16}
\end{equation}
(25)
where \(\varepsilon>0\) to be specified later.
Multiplying \((12)_{1}\) by \(u\) and taking derivative of (25), we obtain
\begin{eqnarray}
\mathcal{K}'(t)&= &H'(t)+\varepsilon\Vert u_{t}\Vert_{2}^{2}+\varepsilon\int_{\Omega}\nabla u\int^{t}_{0}\varpi(t-q) \nabla u(q)dqdx \notag\\
&&-\varepsilon\Vert \nabla u\Vert_{2}^{2}+\varepsilon b\int_{\Omega}\vert u\vert^{p}dx -\varepsilon\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert uy(x, 1, q, t)dqdx. \label{sys2.18}
\end{eqnarray}
(26)
Using
\begin{eqnarray}\label{sys2.19}
\varepsilon\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert uy(x, 1, q, t)dqdx&\leq&\varepsilon\Big\{\delta_{1}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)\Vert u\Vert_{2}^{2}+\frac{1}{4\delta_{1}}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dqdx\Big \},\notag\\
\end{eqnarray}
(27)
and
\begin{eqnarray}
\varepsilon\int_{0}^{t}\varpi(t-q)dq\int_{\Omega}\nabla u \nabla u(q)dxdq&=&\varepsilon\int_{0}^{t}\varpi(t-q)dq\int_{\Omega}\nabla u (\nabla u(q)-\nabla u(t))dxdq+\varepsilon\int_{0}^{t}\varpi(q)dq\Vert \nabla u\Vert_{2}^{2}\notag\\
&\geq & \frac{\varepsilon}{2}\int_{0}^{t}\varpi(q)dq\Vert \nabla u\Vert_{2}^{2}-\frac{\varepsilon}{2}(\varpi o\nabla u). \label{sys2.20}
\end{eqnarray}
(28)
We obtain, from (26),
\begin{eqnarray}
\mathcal{K}'(t)&\geq &H'(t)+\varepsilon\Vert u_{t}\Vert_{2}^{2}-\varepsilon\Big(1-\frac{1}{2}\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2} +\varepsilon b\Vert u\Vert^{p}_{p} \notag\\
&&-\varepsilon\delta_{1}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)\Vert u\Vert_{2}^{2}-\frac{\varepsilon}{4\delta_{1}}\int_{\Omega}\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert y^{2}(x, 1, q, t)dqdx+\frac{\varepsilon}{2}(\varpi o\nabla u). \label{sys2.21}
\end{eqnarray}
(29)
Therefore, using (23) and by setting \(\delta_{1}\) so that, \(\dfrac{1}{4\delta_{1}c_{1}}=\kappa \), substituting in (29), we get
\begin{eqnarray}
\mathcal{K}'(t)&\geq &[1-\varepsilon\kappa]H'(t)+\varepsilon\Vert u_{t}\Vert_{2}^{2}-\varepsilon\Big[\Big(1-\frac{1}{2}\int_{0}^{t}\varpi(q)dq\Big)\Big]\Vert \nabla u\Vert_{2}^{2}+\varepsilon b\Vert u\Vert^{p}_{p} \notag\\
&& -\frac{\varepsilon}{4c_{1}\kappa}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)\Vert u\Vert_{2}^{2}+\frac{\varepsilon}{2}(\varpi o\nabla u). \label{sys2.22}
\end{eqnarray}
(30)
For \(0< a< 1\), from (21)
\begin{eqnarray}
\varepsilon b\Vert u\Vert^{p}_{p} &=&\varepsilon p(1-a)H(t)+\frac{\varepsilon p(1-a)}{2}\Vert u_{t}\Vert_{2}^{2}+\varepsilon ba\Vert u\Vert_{p}^{p}\notag\\
&&+\frac{\varepsilon p(1-a)}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}+\frac{\varepsilon}{2}p(1-a)(\varpi o\nabla u)\notag\\
&&+\frac{\varepsilon p(1-a)}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\label{sys2.23}
\end{eqnarray}
(31)
Substituting in (30), we get
\begin{eqnarray}
\mathcal{K}'(t)&\geq &[1-\varepsilon\kappa]H'(t)+\varepsilon\Big[\frac{p(1-a)}{2}+1\Big]\Vert u_{t}\Vert_{2}^{2}\notag\\
&&+\varepsilon\Big[\Big(\frac{p(1-a)}{2}\Big)\Big(1-\int_{0}^{t}\varpi(q)dq\Big)-\Big(1-\frac{1}{2}\int_{0}^{t}\varpi(q)dq \Big)\Big]\Vert \nabla u\Vert_{2}^{2} \notag\\
&&-\frac{\varepsilon}{4c_{1}\kappa}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)\Vert u\Vert_{2}^{2}+\varepsilon p(1-a)H(t)+\varepsilon ba\Vert u\Vert_{p}^{p}\notag\\
&&+\frac{\varepsilon p(1-a)}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx+\frac{\varepsilon}{2}(p(1-a)+1)(\varpi o\nabla u). \label{sys2.24}
\end{eqnarray}
(32)
Using Poincare’s inequality, we obtain
\begin{eqnarray}
\mathcal{K}'(t)&\geq &[1-\varepsilon\kappa]H'(t)+\varepsilon\Big[\frac{p(1-a)}{2}+1\Big]\Vert u_{t}\Vert_{2}^{2}+\frac{\varepsilon}{2}(p(1-a)+1)(\varpi o\nabla u)\notag\\
&&+\varepsilon\Big\{\Big(\frac{p(1-a)}{2}-1\Big)-\int_{0}^{t}\varpi(q)dq(\frac{p(1-a)-1}{2})\notag\\
&&- \frac{c}{4c_{1}\kappa}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)\Big\}\Vert \nabla u\Vert_{2}^{2} +\varepsilon ab\Vert u\Vert^{p}_{p}+\varepsilon p(1-a)H(t)\notag\\
&&+\frac{\varepsilon p(1-a)}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx. \label{sys2.27}
\end{eqnarray}
(33)
At this point, we choose \(a>0\) so small that
\begin{equation*}
\alpha_{1}=\frac{p(1-a)}{2}-1>0,
\end{equation*}
and assume
\begin{equation}
\int_{0}^{\infty}\varpi(q)dq< \dfrac{\dfrac{p(1-a)}{2}-1}{\Big(\dfrac{p(1-a)}{2}-\dfrac{1}{2}\Big)}=\dfrac{2\alpha_{1}}{2\alpha_{1}+1},\label{c1.1}
\end{equation}
(34)
then we choose \(\kappa\) so large that
\begin{equation*}
\alpha_{2}=\Big(\frac{p(1-a)}{2}-1\Big)-\int_{0}^{t}\varpi(q)dq\Big(\frac{p(1-a)-1}{2}\Big)- \frac{c}{4c_{1}\kappa}\Big(\int_{\tau_{1}}^{\tau_{2}}\vert \mu_{2}(q)\vert dq\Big)>0.
\end{equation*}
Once \(\kappa\) and \(a\) are fixed, we pick \(\varepsilon\) so small enough so that
\begin{equation*}
\alpha_{4}=1-\varepsilon\kappa>0,
\end{equation*}
and
\begin{equation}
\mathcal{K}(t)\leq \frac{b}{p}\Vert u \Vert^p_p. \label{m1.22}
\end{equation}
(35)
Thus, for some \(\beta>0\), estimate (33) becomes
\begin{eqnarray}
\mathcal{K}'(t)&\geq &\beta\Big\{H(t)+\Vert u_{t}\Vert_{2}^{2}
+\Vert \nabla u\Vert_{2}^{2} +(\varpi o\nabla u)+\Vert u\Vert_{p}^{p}+\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx\Big\}, \label{sys2.28}
\end{eqnarray}
(36)
and
\begin{equation}
\mathcal{K}(t)\geq\mathcal{K}(0)>0, \ t>0.\label{sys2.29}
\end{equation}
(37)
Next, using Young’s and Poincare’s inequalities, from (25) we have
\begin{eqnarray}
\mathcal{K}(t)&=&\Big(H+\varepsilon \int_{\Omega}uu_{t}dx+\frac{\varepsilon\mu_{1}}{2}\int_{\Omega}u^{2}dx+\frac{\varepsilon\omega}{2}\int_{\Omega}\nabla u^{2}dx\Big)\notag\\
&\leq&c[H(t)+\vert\int_{\Omega}uu_{t}dx\vert+\Vert u\Vert_{2}^{2}+\Vert \nabla u\Vert_{2}^{2}]\leq c[H(t)+\Vert \nabla u\Vert_{2}^{2}+\Vert u_{t}\Vert_{2}^{2}]. \label{sys2.33}
\end{eqnarray}
(38)
For some \(c>0\). Since, \(H(t) > 0\), we have from (12)
\begin{eqnarray}
&&-\frac{1}{2}\Vert u_{t}\Vert_{2}^{2}-\frac{1}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}-\frac{1}{2}(\varpi o\nabla u)\label{sys2.1.0}\\
&&-\frac{1}{2}\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx+\frac{b}{p}\Vert u\Vert_{p}^{p}>0,\notag
\end{eqnarray}
(39)
then
\begin{eqnarray}
\frac{1}{2}\Big(1-\int_{0}^{t}\varpi(q)dq\Big)\Vert \nabla u\Vert_{2}^{2}&<&\frac{b}{p}\Vert u\Vert_{p}^{p}< \frac{b}{p}\Vert u\Vert_{p}^{p}+(\varpi o\nabla u)\label{sys2.1.1}+\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\notag
\end{eqnarray}
In the other hand, using (7), to get
\begin{eqnarray}
\frac{1}{2}(1-l)\Vert \nabla u\Vert_{2}^{2}&<&\frac{b}{p}\Vert u\Vert_{p}^{p}< \frac{b}{p}\Vert u\Vert_{p}^{p}+(\varpi o\nabla u)+\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\label{sys2.1.2}
\end{eqnarray}
(40)
Consequently,
\begin{eqnarray}
\Vert \nabla u\Vert_{2}^{2}&<&\frac{2b}{p}\Vert u\Vert_{p}^{p}+2(\varpi o\nabla u)+l\Vert \nabla u\Vert_{2}^{2}+2\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx.\label{sys2.1.3}
\end{eqnarray}
(41)
Inserting (41) into (38), to see that there exists a positive constant \(k_{1}\) such that
\begin{eqnarray}
\mathcal{K}(t)&\leq & k_{1}[H(t)+\Vert \nabla u\Vert_{2}^{2}+\Vert u_{t}\Vert_{2}^{2}+\frac{b}{p}\Vert u\Vert_{p}^{p}+(\varpi o\nabla u)(t)\notag\\
&&+\int_{\Omega}\int_{0}^{1}\int_{\tau_{1}}^{\tau_{2}}q\vert \mu_{2}(q)\vert y^{2}(x, \rho, q, t)dqd\rho dx], \forall t>0. \label{sys2.1.4}
\end{eqnarray}
(42)
From inequalities (36) and (42) we obtain the differential inequality
\begin{equation}
\mathcal{K}'(t)\geq \lambda \mathcal{K}(t), \label{sys2.34}
\end{equation}
(43)
where \(\lambda>0\), depending only on \(\beta\) and \(k_{1}\).
A simple integration of (43), we obtain
\begin{equation}
\mathcal{K}(t)\geq \mathcal{K}(0)e^{(\lambda t)}, \forall t>0.\label{m2.0}
\end{equation}
(44)
From (24) and (35), we have
\begin{equation}
\mathcal{K}(t)\leq H(t)\leq\frac{b}{p}\Vert u\Vert_{p}^{p}.\label{m2.1}
\end{equation}
(45)
By (44) and (45), we have
\begin{equation*}
\Vert u\Vert_{p}^{p}\geq C e^{(\lambda t)}, \forall t>0.
\end{equation*}
Therefore, we conclude that the solution in the \(L_{p}\)-norm growths exponentially.
This completes the proof.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
