1. Introduction
Consider the following problem:
\begin{align}
&u_{tt}-div\left( \frac{\mid \nabla u\mid ^{2m-2}\nabla u}{\sqrt{1+\mid
\nabla u\mid ^{2m}}}\right) -\omega \Delta u_{t}+\mu u_{t}=u\mid u\mid
^{p-2},\ x\in \Omega ,\ t\geq 0, \label{1}
\end{align}
(1)
\begin{align}
&u\left( x,0\right) =u_{0}\left( x\right) ,\ u_{t}\left( x,0\right)
=u_{1}\left( x\right) , \label{2}
\end{align}
(2)
\begin{align}
&u\left( x,t\right) =0,\ x\in \partial \Omega ,\ t\geq 0, \label{3}
\end{align}
(3)
where \(\Omega \) is a bounded regular domain in \(\mathbb{R}^{n},\) \(n\geq 1\) with a smooth boundary \(\partial \Omega \). \(\omega \), \(\mu \)
and \(m\), \(p\) are real numbers.
The nonlinear wave equations
\begin{align}
&u_{tt}-\Delta u-\omega \Delta u_{t}+\mu u_{t}=u\mid u\mid ^{p-2},\ x\in
\Omega ,\ t\geq 0, \label{4}
\end{align}
(4)
\begin{align}
&u\left( x,0\right) =u_{0}\left( x\right) ,\ u_{t}\left( x,0\right)
=u_{1}\left( x\right) ,\ t\geq 0, \label{5}
\end{align}
(5)
\begin{align}
&u\left( x,t\right) =0,x\in \partial \Omega ,t\geq 0, \label{6}
\end{align}
(6)
has been investigated by many authors [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10]. In the absence of the nonlinear source term,
it is well know that the presence of one damping term ensures global
existence and decay of solutions for arbitrary initial condition
[
5,
6]. For \(\omega =\mu =0\) the nolinear term \(u\mid
u\mid ^{p-2}\) causes finite time blow up of solutions with negative energy
[
2]. The interaction between the damping and the source
terms was first considered by Levine [
11]. He showed that
solutions with negative initial energy blows up in finite time. When \(\omega
=0\) and the linear term \(u_{t}\) is replaced by \(\mid u_{t}\mid ^{r-2}u_{t}\),
Georgiev and Todorowa [
12] extended Levin’s result to the case
where \(r>2\). In their work, the authors introduced a method different from
the one know as the concavity method. The termined suitable relations
between \(r\) and \(p\), for whith there is global existence or alternatively
fnite time blow-up.
For the initial boundary value problem of a quasilinear equation
\begin{equation*}
u_{tt}-div\left( {\mid \nabla u\mid ^{m-2}\nabla u}\right) +au_{t}\mid
u_{t}\mid ^{p-2}-\Delta u_{t}=bu\mid u\mid ^{r-2},
\end{equation*}
\begin{align}
&x\in \Omega ,\ t\geq 0,
\label{7}
\end{align}
(7)
\begin{align}
&u\left( x,0\right) =u_{0}\left( x\right) \in W_{0}^{1,m}\left( \Omega
\right) ,\ u_{t}\left( x,0\right) =u_{1}\left( x\right) \in L^{2}\left(
\Omega \right) ,\ t\geq 0, \label{8}
\end{align}
(8)
\begin{align}
&u\left( x,t\right) =0,\ x\in \partial \Omega ,\ t\geq 0. \label{9}
\end{align}
(9)
Yang and Chen [
13,
14] studied the problem (7)-(9) and obtained global existence results under the growth
assumptions on the nonlinear terms and the initial value. This global
existence results have been improved by Liu and Zhao [
15] by
using a new method. In [
13], the author considered a similar
problem to (7)-(9) and proved a blow-up result under the
condition \(p>max(r,m)\) and the energy is sufficiently negative. Messaoudi
and Said-Houari [
16] improved the results in [
15]
and showed that blow-up takes place for negative initial data only
regardless of the size of \(\Omega\) Messaoudi in [
17] showed
that for \(m=2\), the decay is exponential. In absence of strong damping \(-\Delta u_{t}\) equation (7) becames
\begin{equation}
\ u_{tt}-div\left( {\mid \nabla u\mid ^{m-2}\nabla u}\right) +au_{t}\mid
u_{t}\mid ^{p-2}=bu\mid u\mid ^{r-2},\ x\in \Omega ,\ t\geq 0. \label{10}
\end{equation}
(10)
For \(b=0\), it is well known that the damping term assures global existence
and decay of the solution energy for arbitrary initial value [
18]. For \(a=0\), the source term causes finite time blow-up of solutions with
negative initial energy if \(r>m\) (see [
2]). When the
quasilinear operator \(-div\left( {\mid \nabla u\mid ^{m-2}\nabla u}\right) \)
is replaced by \(\Delta ^{2}u\) , Wu and Tsai [
19] showed that
the solution is global in time under some conditions without the relation
between \(p\) and \(r\). They also proved that the local solution blows up
infinite time if \(r>p\) and the initial energy is nonnegative, and gave the
decay estimates of the energy function and the lifespan of solutions. In
this paper, we show that the local solutions of the problem (1)-(3) can be extented in infinite time to global solutions with the
some conditions on initial data in the stable set for which the solutions
decay expontially with \(L_{p}\) norm. The key tool in the proof is an idea of
Haraux and Zuazua [
6] and [
9] with is based on
the construction of a suitable Lyapunov function.
2. Assumptions and preliminaries
In this section, we present some material needed in the proof in our result.
Lemma 1. \(\left( Young^{\prime }s\ inequality\right) \) \(Let\ a,\ b\geq 0\)
and \(\frac{1}{p}+\frac{1}{q}\)=1 for \( 1
< p, q 0 \) is an arbitrary constant, and \(C\left( \delta \right) \)
is a positive constant depending on \(\delta \).
Lemma 2.
Let s be a number with \(2\leq s r\). Then there is a constant \(C\)
depending on \(\Omega \) and s such that \(\left\Vert u\right\Vert _{s}\) \(\leq C\left\Vert \nabla u\right\Vert _{r}\), \(u\in W_{0}^{1,r}\left( \Omega \right) .\)
We denote the total energy related to the problem (1)-(3) by
\begin{equation}
E\left( t\right) =\frac{1}{2}\left\Vert u_{t}\right\Vert _{2}^{2}+\frac{1}{m}
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}dx-
\frac{1}{p}\left\Vert u\right\Vert _{p}^{p}. \label{11}
\end{equation}
(11)
We also introduce the following functionals:
\begin{equation}
I\left( t\right) =\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u\right\vert ^{2m}}dx-\left\Vert u\right\Vert _{p}^{p}, \label{12}
\end{equation}
(12)
\begin{equation}
J\left( t\right) =\frac{1}{m}\underset{\Omega }{\int }\sqrt{1+\left\vert
\nabla u\right\vert ^{2m}}dx-\frac{1}{p}\left\Vert u\right\Vert _{p}^{p}.
\label{13.}
\end{equation}
(13)
As in [
20], we can now define the so called ” Nehari manifold”
as follows:
\begin{equation*}
\mathcal{N}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) \backslash
\left\{ 0\right\} ;\text{ }I\left( t\right) =0\right\} .
\end{equation*}
\(\mathcal{N}\) separates the two unbounded sets:
\begin{equation*}
\mathcal{N}^{+}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) ;\text{
}I\left( t\right) >0\right\} \cup \left\{ 0\right\}
\end{equation*}
and
\begin{equation*}
\mathcal{N}^{-}\text{=}\left\{ u\in W_{0}^{1,m}\left( \Omega \right) ;\text{
}I\left( t\right) < 0\right\} .
\end{equation*}
Assumptions:
- \(\left( A1\right) :\) Assume that \(I\left( 0\right) >0,\) and \(0< E\left(
0\right) \) such that
\begin{equation}
B=c^{p}\left( \frac{mp}{p-m}E\left( 0\right) \right) ^{\frac{p-m}{m}}< 1.%
\text{ } \label{14}
\end{equation}
(14)
where \(c\) is the Poincaré constant.
- \(\left( A2\right) :\) \(p\) satisfies
\[(2< m< p\leq\dfrac{nm}{n-m}, n\geq m; 2< m < p\leq+\infty, n < m.\]
For simplicity, we define the weak solutions of (1)-(3) over
the interval \(\left[ 0,T\right) \), but it is to be understood throughout
that \(T\) is either infinity or the limit of the existence interval.
Definition 1.
We say that \(u\left( x,t\right) \) is a weak solution of the
problem (1)-(3) on the interval \(\Omega \times \left[
0,T\right) ,\) if \(u\in L^{\infty }\left( \left[ 0,T\right)
;W_{0}^{1,m}\left( \Omega \right) \right) ,u_{t}\ \in L^{\infty }\left( %
\left[ 0,\text{ }T\right) ;\text{ }L^{2}\left( \Omega \right) \right) \cap
L^{2}\left( \left[ 0,\text{ }T\right) ;\ H_{0}^{1}\left( \Omega \right)
\right) \) satisfy the following conditions:
-
\(\left( i\right) \)
\begin{equation}
\left( u^{\prime \prime }\left( t\right) ,\phi \right) +\left( \frac{\mid
\nabla u\mid ^{2m-2}\nabla u}{\sqrt{1+\mid \nabla u\mid ^{2m}}},\nabla \phi
\right) +\omega \left( \nabla u^{\prime },\nabla \phi \right) +\mu \left(
u^{\prime },\phi \right) =\left( u\lvert u\rvert ^{p-2},\phi \right) ,
\label{15}
\end{equation}
(15)
for any function \(\phi \in W_{0}^{1,m}\left( \Omega \right) \) and a.e. \(t\in
\left[ 0,T\right)\).
-
\(\left( ii\right) \)
\begin{equation}
u\left( x,0\right) =u_{0}\left( x\right) \in L^{2}\left( \Omega \right) ,\
u_{t}\left( x,0\right) =u_{1}\left( x\right) \in L^{1}\left( \Omega \right) .
\label{16}
\end{equation}
(16)
Theorem 1.
(Local existence) Suppose that \(u_{0}\in L^{2}\left(
\Omega \right) ,\ u_{1}\in L^{1}\left( \Omega \right) \) and \(E\left(
0\right) >0,\) then there exists \(T>0\) such that problem (1)-(3) has a unique solution u satisfying \(u\ \in L^{\infty}\left( [0, T ];\ W_{0}^{1,m}\left( \Omega\right)\right), \) \(u_{t}\ \in L^{\infty }\left( \left[ 0,\text{ }T\right) ;\text{ }L^{2}\left( \Omega \right) \right) \cap L^{2}\left( \left[ 0,\text{ }T\right) ;\ H_{0}^{1}\left( \Omega \right) \right) .\)
3. Global existence and exponential decay of solutions
In this section we are going to obtain the existence of local
solutions to the problem (1)-(3) and exponential
decay of solution. We will use the Faedo- Galerkin’s method approximation.
Let \(\{w_{l}\}_{l=1}^{\infty }\) be a basis of \(W_{0}^{1,m}\left( \Omega
\right) \) wich constructs a complete orthonormal system in \(L^{2}\left(
\Omega \right) .\) Denote by \(V_{k}=span\{w_{1},w_{2},…,w_{k}\}\) the
subspace generated by the first \(k\) vectors of the basis \(
\{w_{l}\}_{l=1}^{\infty }.\) By the normalization, we have \(\lVert
w_{l}\rVert =1.\) for any given integer k, we consider the approximation
solution
\begin{equation*}
u_{k}\left( t\right) =\sum_{l=1}^{k}u_{lk}\left( t\right) v_{l},
\end{equation*}
where \(u_{k}\) is the solutions to the following Cauchy problem
\begin{equation}
\left( u_{k}^{\prime \prime }\left( t\right) ,v_{l}\right) +\left( \frac{\mid \nabla u_{k}\mid ^{2m-2}\nabla u_{k}}{\sqrt{1+\mid \nabla u_{k}\mid
^{2m}}},\nabla v_{l}\right) +\omega \left( \nabla u_{k}^{\prime },\nabla
v_{l}\right) +\mu \left( u_{k}^{\prime },v_{l}\right) =\left( u_{k}\lvert
u_{k}\rvert ^{p-2},v_{l}\right) , \label{17}
\end{equation}
(17)
where \(l=1,…,k,\) with initial conditions \(u_{k}\left( 0\right) =u_{0k}\)
and \(u_{k}^{\prime }\left( 0\right) =u_{1k},\) \(u_{k}\left( 0\right) \) and \(
u_{k}^{\prime }\left( 0\right) \) are chosen in \(V_{k}\) such that
\begin{equation}
\sum_{l=1}^{k}\left( u_{0},v_{l}\right) v_{l}=u_{0k}\longrightarrow u_{0}\
in\ L^{2}\left( \Omega \right) ;\sum_{l=1}^{k}\left( u_{1},v_{l}\right)
v_{l}=u_{1k}\longrightarrow u_{1}\ in\ L^{1}\left( \Omega \right) .
\label{18}
\end{equation}
(18)
Well known results on the solvability of nonlinear ODE provide the existence
of a solution to problem (17)-(18) on interval \(\left[ 0,\tau\right) \) for some \(\tau >0\) and we can extend this solution to the whole
interval \(\left[ 0,T\right] \) for any given \(T>0\) by making use of the a
priori estimates below. Multiplying equation (17) by \(u_{lk}^{\prime }\left( t\right) \) and
sum for \(l=1,…,k,\) we obtain
\begin{equation}
\frac{d}{dt}\left( \frac{1}{2}\left\Vert u_{k}^{\prime }\right\Vert _{2}^{2}+
\frac{1}{m}\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u_{k}\right\vert ^{2m}}dx-\frac{1}{p}\left\Vert u_{k}\right\Vert
_{p}^{p}\right)
=-\left( \omega \underset{\Omega }{\int }\lvert \nabla u_{k}\rvert
_{2}^{2}dx+\mu \underset{\Omega }{\int }\lvert u_{k}^{\prime }\rvert
_{2}^{2}dx\right). \label{19}
\end{equation}
(19)
Integrating (19) over \(\left( 0,t\right)\), we
obtain the estimate
\begin{equation}
\frac{1}{2}\left\Vert u_{k}^{\prime }\right\Vert _{2}^{2}+\frac{1}{m}
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx-
\frac{1}{p}\left\Vert u_{k}\right\Vert _{p}^{p}
+\omega \int_{0}^{t}\underset{\Omega }{\int }\lvert \nabla u_{k}\rvert
_{2}^{2}dx+\mu \int_{0}^{t}\underset{\Omega }{\int }\lvert u_{k}^{\prime
}\rvert _{2}^{2}dx\leq E\left( 0\right) . \label{20}
\end{equation}
(20)
Since \(I\left( 0\right) >0,\) then there exists \(\tau < T\) by
continuity such that \(I\left( t\right) \geq 0,\). We get from (12) and (13) that
\begin{equation}
J\left( u_{k}\left( t\right) \right) =\frac{p-m}{mp}\underset{\Omega }{\int }%
\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx+\frac{1}{p}I\left(
u_{k}\left( t\right) \right) \label{21}
\end{equation}
(21)
\begin{equation}
J\left( u_{k}\left( t\right) \right) \geq \frac{p-m}{mp}\underset{\Omega }{%
\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx, \forall
t\in \left[ 0,\text{ }\tau \right] . \label{22}
\end{equation}
(22)
Hence we have
\begin{equation}
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}
dx\leq \frac{mp}{p-m}J\left( u_{k}\left( t\right) \right) . \label{23}
\end{equation}
(23)
From (11) and (13) we obvioulsy have \(\forall
t\in \left[ 0,\text{ }\tau \right] ,\) \(J\left( u_{k}\left( t\right) \right)
\leq E\left( u_{k}\left( t\right) \right) .\) Thus we obtain
\begin{equation}
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}
dx\leq \frac{mp}{p-m}E\left( u_{k}\left( t\right) \right) . \label{24}
\end{equation}
(24)
Since \(E\) is a decreasing function of \(t,\) we have
\begin{equation}
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}
dx\leq \frac{mp}{p-m}E\left( 0\right) , \forall t\in \left[ 0,\text{
}\tau \right] \label{25}
\end{equation}
(25)
By using Lemma 2, we easily have
\begin{eqnarray*}
\left\Vert u_{k}\right\Vert _{p}^{p} &\leq &c^{p}\left\Vert \nabla
u_{k}\right\Vert _{m}^{p}=c^{p}\left( \underset{\Omega }{\int }\left\vert
\nabla u_{k}\right\vert ^{m}dx\right) ^{\frac{p}{m}}\leq c^{p}\left(
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}
dx\right) ^{\frac{p}{m}} \\
&\leq &c^{p}\left( \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u_{k}\right\vert ^{2m}}dx\right) ^{\frac{p-m}{m}}\underset{\Omega }{\int }
\sqrt{1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx
\end{eqnarray*}
Using the inequality (25), we deduce
\begin{equation*}
\left\Vert u_{k}\right\Vert _{p}^{p}\leq c^{p}\left( \frac{mp}{p-m}E\left(
0\right) \right) ^{\frac{p-m}{m}}\underset{\Omega }{\int }\sqrt{1+\left\vert
\nabla u_{k}\right\vert ^{2m}}dx.
\end{equation*}
Now exploiting the inequality (14), we obtain
\begin{equation}
\left\Vert u_{k}\right\Vert _{p}^{p}\leq \underset{\Omega }{\int }\sqrt{
1+\left\vert \nabla u_{k}\right\vert ^{2m}}dx. \label{26}
\end{equation}
(26)
Hence \(\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert
^{2m}}dx-\) \(\left\Vert u_{k}\right\Vert _{p}^{p}>0,\) \(\forall t\in \left[ 0,
\text{ }\tau \right] ,\) this shows that \(I\left( u_{k}\left( t\right)
\right) >0,\) by repeating this procedure, \(\tau \) is extended to T.
Since \(\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u_{k}\right\vert
^{2m}}dx>\lVert \nabla u_{k}\rVert _{m}^{m},\) it follows from (20) and (26) that
\begin{equation}
\frac{1}{2}\left\Vert u_{k}^{\prime }\right\Vert _{2}^{2}+\frac{p-m}{pm}%
\lVert \nabla u_{k}\rVert _{m}^{m}+
+\omega \int_{0}^{t}\underset{\Omega }{\int }\lvert \nabla u_{k}\rvert
_{2}^{2}dx+\mu \int_{0}^{t}\underset{\Omega }{\int }\lvert u_{k}^{\prime
}\rvert _{2}^{2}dx\leq E\left( 0\right) . \label{27}
\end{equation}
(27)
From (27), we have
\begin{equation}
\left\{
\begin{array}{c}
\left\{ u_{k}\right\} \mathit{\ }\text{is uniformly bounded
in }L^{\infty }\left( \left[ 0,T\right] ;W_{0}^{1,m}\left( \Omega \right)
\right) , \\
\left\{ u_{k}\right\} \rightharpoonup u\mathit{\ }\text{is
uniformly bounded in }L^{2}\left( \left[ 0,T\right] ;H_{0}^{1}\left(
\Omega \right) \right) , \\
\left\{ u_{k}^{\prime }\right\} \mathit{\ }\text{is
uniformly bounded in }L^{\infty }\left( \left[ 0,T\right] ;L^{2}\left(
\Omega \right) \right) , \\
\left\{ u_{k}^{\prime }\right\} \mathit{\ }\text{is
uniformly bounded in }L^{2}\left( \left[ 0,T\right] ;L^{2}\left( \Omega
\right) \right).
\end{array}
\right. \label{28}
\end{equation}
(28)
Furthermore, we have from Lemma 2 and (28) that
\begin{equation}
\left\{ \lvert u_{k}\rvert ^{p}u_{k}\right\} \mathit{\ }\text{
is uniformly bounded in }L^{\infty }\left( \left[ 0,T\right]
;L^{2}\left( \Omega \right) \right) . \label{29}
\end{equation}
(29)
By (28) and (29), we infer that there exists a subsequence
of \(u_{k}\) (denote still by the same symbol) and a function \(u\) such that
\begin{equation}
\left\{
\begin{array}{c}
u_{k}\rightharpoonup u\mathit{\ }\text{weakly star in }
L^{\infty }\left( \left[ 0,T\right] ;W_{0}^{1,m}\left( \Omega \right)
\right) , \\
u_{k}\rightharpoonup u\mathit{\ }\text{weakly star in }
L^{2}\left( \left[ 0,T\right] ;H_{0}^{1}\left( \Omega \right) \right) , \\
u_{k}^{\prime }\rightharpoonup u^{\prime }\mathit{\ }\text{
weakly star in }L^{\infty }\left( \left[ 0,T\right] ;L^{2}\left( \Omega
\right) \right) , \\
u_{k}^{\prime }\rightharpoonup u^{\prime }\mathit{\ }\text{
weakly star in }L^{2}\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right)
\right) , \\
\lvert u_{k}\rvert ^{p-2}u_{k}\rightharpoonup \mathcal{X}\mathit{\ }\text{
weakly star in }L^{\infty }\left( \left[ 0,T\right]
;L^{2}\left( \Omega \right) \right) .
\end{array}
\right. \label{30}
\end{equation}
(30)
By the Aubin-Lions compactness Lemma [
7], we conclude from (30) that
\begin{equation*}
\left\{
\begin{array}{c}
u_{k}\rightharpoonup u\ \ \text{strongly in}\ \ C\left( \left[
0,T\right] ;L^{2}\left( \Omega \right) \right) , \\
u_{k}^{\prime }\rightharpoonup u^{\prime }\mathit{\ }\text{
strongly in }C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right)
\right) ,
\end{array}
\right.
\end{equation*}
and
\begin{equation}
u_{k}\rightharpoonup u\mathit{\ }\text{almost everywhere in }\left[ 0,T\right] \times \Omega . \label{31}
\end{equation}
(31)
It follows from Lemma 1.3 in [
21] and (31)
\begin{equation}
\lvert u_{k}\rvert ^{p-2}u_{k}\rightharpoonup \lvert u\rvert ^{p-2}u\mathit{
\ }\text{weakly star in }L^{\infty }\left( \left[ 0,T
\right] ;L^{2}\left( \Omega \right) \right) . \label{32}
\end{equation}
(32)
By the last formula (32) and (30), we obtain \(\mathcal{X}=\lvert u\rvert ^{p-2}u\)
On the other hand, taking \(\phi =1\), (17) become
\begin{equation}
\left( u_{k}^{\prime \prime }\left( t\right) ,1\right) +\mu \left(
u_{k}^{\prime },1\right) =\left( u_{k}\lvert u_{k}\rvert ^{p-2},1\right).
\label{33}
\end{equation}
(33)
We have
\begin{equation*}
\lvert \left( u_{k}^{\prime \prime }\left( t\right) ,1\right) +\mu \left(
u_{k}^{\prime },1\right) \rvert \geq \lVert u_{k}^{\prime \prime }\rVert
-\mu \lVert u_{k}^{\prime }\rVert.
\end{equation*}
Since, the measure of \(\Omega \) is finite, by the embedding theorem,
(30) and (33), we obtain
\begin{equation*}
\lVert u_{k}^{\prime \prime }\rVert \leq C,
\end{equation*}
then
\begin{equation*}
\left\{ u_{k}^{\prime \prime }\right\} \mathit{\ }\text{is
uniformly bounded in }L^{\infty }\left( \left[ 0,T\right] ;L^{1}\left(
\Omega \right) \right) .
\end{equation*}
Similarly, we have
\begin{equation}
u_{k}^{\prime \prime }\rightharpoonup u^{\prime \prime }\mathit{\ }\text{
weakly star in }L^{\infty }\left( \left[ 0,T\right]
;L^{1}\left( \Omega \right) \right) , \label{34}
\end{equation}
(34)
Setting up \(k\longrightarrow \infty \) and passing to the limit in (17), we obtain
\begin{equation*}
\left( u^{\prime \prime }\left( t\right) ,v_{l}\right) +\left( \frac{\mid
\nabla u\mid ^{2m-2}\nabla u}{\sqrt{1+\mid \nabla u_{k}\mid ^{2m}}},\nabla
v_{l}\right) +\omega \left( \nabla u^{\prime },\nabla v_{l}\right) +\mu
\left( u^{\prime },v_{l}\right) =\left( u\lvert u\rvert ^{p-2},v_{l}\right) ,
\end{equation*}
\(l=1,…,k.\) Since \(\{v_{l}\}_{l=1}^{\infty }\) is a base of \(W_{0}^{1,m}\left( \Omega \right)\), we deduce that \(u\) satisfies (1).
From (30), (34) and Lemma 3.1.7 in [22],
with \(B=L^{2}\left( \Omega \right) \) and \(B=L^{1}\left( \Omega \right) ,\)
respectively, we infer that
\begin{equation}
\left\{
\begin{array}{c}
u_{k}\left( 0\right) \rightharpoonup u\left( 0\right) \ \text{ weakly in}
\text{ }\ L^{2}\left( \Omega \right) , \\
u_{k}^{^{\prime }}\left( 0\right) \rightharpoonup u^{^{\prime }}\left(
0\right) \ \text{weakly star in}\ L^{1}\left( \Omega \right) .
\end{array}%
\right. \label{35}
\end{equation}
(35)
We get from (18) and (35) that \(u\left( 0\right) =u_{0},\) \(u^{\prime }\left( 0\right) =u_{1}.\)
Thus, the proof is complete.
Lemma 3.
Assume that \(p>m\) and \( u_{0}\in \mathcal{N}^{+},\) \(
u_{1}\in L^{2}\left( \Omega \right) .\) If \(0< E\left( 0\right) \) and
satisfy (14) then the local solution of the problem (1)-(3) is global in time.
Proof.
Since the map \(t\longmapsto E\left( t\right) \) is a decreasing of
the time \(t,\) we have
\begin{equation}
E\left( 0\right) \geq E\left( t\right) =\frac{1}{2}\left\Vert
u_{t}\right\Vert _{2}^{2}+\frac{p-m}{mp}\underset{\Omega }{\int }\sqrt{%
1+\left\vert \nabla u\right\vert ^{2m}}dx+\frac{1}{p}I\left( t\right)
\label{36}
\end{equation}
(36)
which give
\begin{equation}
E\left( 0\right) \geq E\left( t\right) \geq \frac{1}{2}\left\Vert
u_{t}\right\Vert _{2}^{2}+\frac{p-m}{mp}\underset{\Omega }{\int }\sqrt{%
1+\left\vert \nabla u\right\vert ^{2m}}dx \label{37}
\end{equation}
(37)
thus, \(\forall t\in \left[ 0,\text{ }T\right) ,\) \(\left\Vert
u_{t}\right\Vert _{2}^{2}+\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u\right\vert ^{2m}}dx\) is uniformly bounded by a constant depending only on
\(E\left( 0\right) ,\) \(p\) and \(m\) then the solution is global, so \(T_{\max
}=\infty .\)
Theorem 2.
Assume that \(p>m\). Let \(u_{0}\in
\mathcal{N}^{+}\) and \(u_{1}\in L^{2}\left( \Omega \right) .\)
Moreover, assume that \(0< E\left( 0\right) \) and satisfy (14). Then there exists two positive constants \(
\alpha \) and \(\beta \) independent of \(t\) such
that: \(00.\)
Proof.
Since we have proved that \(t\geq 0,\) \(u\left( t\right) \in
\mathcal{N}^{+},\) we already have
\begin{equation*}
00.\)
\begin{equation}
L\left( t\right) =E\left( t\right) +\epsilon \underset{\Omega }{\int }
u_{t}udx+\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2}.
\label{38}
\end{equation}
(38)
We prove that \(L\left( t\right) \) and \(E\left( t\right) \) are equivalent in
the sens that there exist two constants \(B_{1}\) and \(B_{2}\) depending on \(
\epsilon\) such that for \(t\geq 0\)
\begin{equation}
B_{1}E\left( t\right) \leq L\left( t\right) \leq B_{2}E\left( t\right) .
\label{39}
\end{equation}
(39)
By the Lemma 1, we have
\begin{equation*}
L\left( t\right) =E\left( t\right) +\epsilon \underset{\Omega }{\int }
u_{t}udx+\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2}
\leq E\left( t\right) +\epsilon \left( \frac{1}{4\delta }\left\Vert
u_{t}\right\Vert _{2}^{2}+\delta \left\Vert u\right\Vert _{2}^{2}\right) +
\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert _{2}^{2}.
\end{equation*}
Thanks of the Poincaré inequality and since \(\delta\) is an arbitrary
constant, we choose \(\delta \) small suffisant for that,
\begin{equation}
\delta \left\Vert u\right\Vert _{2}^{2}\leq \delta C\left\Vert \nabla
u\right\Vert _{2}^{2}\leq \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u\right\vert ^{2m}}dx \label{40}
\end{equation}
(40)
Then, we get
\begin{equation*}
L\left( t\right) \leq E\left( t\right) +\epsilon \frac{1}{4\delta }
\left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left( \delta C+\frac{\omega }{
2}\right) \left\Vert \nabla u\right\Vert _{2}^{2}
\leq E\left( t\right) +\epsilon \frac{1}{4\delta }\left\Vert
u_{t}\right\Vert _{2}^{2}+\epsilon \left( 1+\frac{\omega }{2}\right)
\underset{\Omega }{\int }\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}dx.
\end{equation*}
By (37), we get
\begin{equation}
L\left( t\right) \leq E\left( t\right) +\epsilon \frac{1}{2\delta }E\left(
t\right) +\epsilon \left( 1+\frac{\omega }{2}\right) \frac{mp}{p-m}E\left(
t\right)
\leq B_{2}E\left( t\right) , \label{41}
\end{equation}
(41)
where \(B_{2}=\left( 1+\epsilon \frac{1}{2\delta }+\epsilon \left( 1+\frac{\omega }{2}\right) \frac{mp}{p-m}\right)\).
On the other hand, we have
\begin{align*}
L\left( t\right) &\geq E\left( t\right) -\epsilon \left( \frac{1}{4\delta }
\left\Vert u_{t}\right\Vert _{2}^{2}+\delta \left\Vert u\right\Vert
_{2}^{2}\right) +\frac{\epsilon \omega }{2}\left\Vert \nabla u\right\Vert
_{2}^{2}\\
&\geq E\left( t\right) -\epsilon \frac{1}{4\delta }\left\Vert
u_{t}\right\Vert _{2}^{2}-\epsilon \delta \left\Vert u\right\Vert _{2}^{2}\\
&\geq E\left( t\right) -\epsilon \frac{1}{2\delta }E\left( t\right) -\epsilon
\delta \left\Vert u\right\Vert _{2}^{2}\\
&\geq \left( 1-\epsilon \frac{1}{2\delta }\right) E\left( t\right) -\epsilon
\delta \left\Vert u\right\Vert _{2}^{2}.
\end{align*}
From (37) and (40), we obtain
\begin{equation}
L\left( t\right) \geq \left( 1-\epsilon \frac{1}{2\delta }-\epsilon \frac{mp%
}{p-m}\right) E\left( t\right) =B_{1}E\left( t\right) , \label{42}
\end{equation}
(42)
where \(B_{1}=\left( 1-\epsilon \frac{1}{2\delta }-\epsilon \frac{mp}{p-m}%
\right)\).
Now, we have
\begin{align*}
\frac{d}{dt}L\left( t\right) &=-\omega \left\Vert \nabla u_{t}\right\Vert
_{2}^{2}-\mu \left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left\Vert
u_{t}\right\Vert _{2}^{2}
+\epsilon \underset{\Omega }{\int }{div}\left( \frac{\left\vert \nabla
u\right\vert ^{2m-2}\nabla u}{\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}}%
\right) udx+\epsilon \left\Vert u\right\Vert _{p}^{p}-\epsilon \mu \underset{%
\Omega }{\int }u_{t}udx\\
&=-\omega \left\Vert \nabla u_{t}\right\Vert _{2}^{2}-\mu \left\Vert
u_{t}\right\Vert _{2}^{2}+\epsilon \left\Vert u_{t}\right\Vert
_{2}^{2}-\epsilon \underset{\Omega }{\int }\frac{\left\vert \nabla
u\right\vert ^{2m}}{\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}}dx
+\epsilon \left\Vert u\right\Vert _{p}^{p}-\epsilon \mu \underset{\Omega }{%
\int }u_{t}udx.
\end{align*}
So that
\begin{equation}
\frac{d}{dt}L\left( t\right) \leq -\omega \left\Vert \nabla u_{t}\right\Vert
_{2}^{2}+\left( \epsilon \left( \frac{\mu }{4\delta }+1\right) -\mu \right)
\left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \mu \delta \left\Vert
u\right\Vert _{2}^{2}
-\epsilon \underset{\Omega }{\int }\frac{\left\vert \nabla u\right\vert ^{2m}
}{\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}}dx+\epsilon \underset{\Omega
}{\int }\sqrt{1+\left\vert \nabla u\right\vert ^{2m}}dx. \label{45}
\end{equation}
(43)
So
\begin{equation}
\frac{d}{dt}L\left( t\right) \leq \left( \epsilon \left( \frac{\mu }{4\delta
}+1\right) -\mu \right) \left\Vert u_{t}\right\Vert _{2}^{2}+\epsilon \left(
1+\mu \right) \underset{\Omega }{\int }\sqrt{1+\left\vert \nabla
u\right\vert ^{2m}}dx. \label{46}
\end{equation}
(44)
Using the inequality (37) and (44), we deduce
\begin{eqnarray*}
\frac{d}{dt}L\left( t\right) &\leq &2\left( \epsilon \left( \frac{\mu }{
4\delta }+1\right) -\mu \right) E\left( t\right) +\epsilon \left( 1+\mu
\right) \frac{mp}{p-m}E\left( t\right) \\
&\leq &-\left( 2\mu -\epsilon \left( \left( \frac{\mu }{2\delta }+2\right)
+\left( 1+\mu \right) \frac{mp}{p-m}\right) \right) E\left( t\right).
\end{eqnarray*}
We choosing \(\epsilon \) small enough such that
\begin{equation}
-\left( 2\mu -\epsilon \left( \left( \frac{\mu }{2\delta }+2\right) +\left(
1+\mu \right) \frac{mp}{p-m}\right) \right) =\zeta < 0. \label{47}
\end{equation}
(45)
So
\begin{equation}
\frac{d}{dt}L\left( t\right) \leq \zeta E\left( t\right). \label{48}
\end{equation}
(46)
From (39), we have
\begin{equation}
\frac{d}{dt}L\left( t\right) \leq \frac{\zeta }{B_{2}}L\left( t\right).
\label{49}
\end{equation}
(47)
Integrating the provious differential inequality (47) between \(0\)
and \(t\) gives the following estimate for the function \(L:\)
\begin{equation}
L\left( t\right) \leq ce^{\frac{\zeta }{B_{2}}t}, \forall t\geq 0.
\label{50}
\end{equation}
(48)
Consequently, by using (39) once again, we conclude
\begin{equation}
E\left( t\right) \leq ke^{\frac{\zeta }{B_{2}}t}, \forall t\geq 0.
\label{51}
\end{equation}
(49)
By using (26) and (37) we easily have
\begin{equation}
\left\Vert u\right\Vert _{p}^{p}\leq k_{1}e^{\frac{\zeta }{B_{2}}t}, \forall t\geq 0 . \label{52}
\end{equation}
(50)
The proof is complete.
4. Conclusion
In this paper, we have studied a class of hyperbolic
equation supplemented with Dirichlet boundary conditions as a model of wave
equation with damping and source nonlinear terms. We showed that the
solution with positive initial energy exponentially decay, this is mainly
due to the presence of one of term of weak or strong damping.
Acknowledgments:
The authors wish to thank deeply the anonymous
referee for useful remarks and careful reading of the proofs presented in
this paper.
Author Contributions:
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflict of Interests:
The authors declare no conflict of interest.
