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On the decay and blow up of solutions for a quasilinear hyperbolic equations with nonlinear damping and source terms
Boundary Value Problems volume 2015, Article number: 127 (2015)
Abstract
In this work we investigate the global existence, decay, and blow up of solutions for a quasilinear hyperbolic equation. We prove the decay estimates of the energy function by using Nakao’s inequality. Also, we obtain the blow up of solutions and lifespan estimates in three different ranges of the initial energy.
1 Introduction
In this work we study the following quasilinear hyperbolic equations:
where Ω is a bounded domain with smooth boundary ∂Ω in \(R^{n}\) (\(n\geq1\)); \(m>0\), \(p,q\geq1\).
Problems of this type arise in physics. For example, this problem represents the longitudinal motion of a viscoelastic configuration which obeys a nonlinear Voight model [1, 2].
When \(m=0\), (1) becomes the following wave equation with nonlinear and strong damping terms:
Gerbi and Houari [3] studied the exponential decay, Chen and Liu [4] studied the global existence, decay, and exponential growth of solutions of the problem (2). Also, Gazzola and Squassina [5] studied the global existence and blow up of solutions of the problem (2), for \(q=1\).
In the absence of the strong damping term \(\bigtriangleup u_{t}\) and \(m=0\), the problem (1) can be reduced to the following wave equation with nonlinear damping and source terms:
Many authors have investigated the local existence, blow up, and asymptotic behavior of solutions of (3); see [6–11]. The interaction between the damping \((\vert u_{t}\vert ^{q-1}u_{t})\) and the source term \((\vert u\vert ^{p-1}u)\) makes the problem more interesting. Levine [7, 8] first studied the interaction between the linear damping (\(q=1\)) and source term by using a concavity method. But this method cannot be applied in the case of a nonlinear damping term. Georgiev and Todorova [6] extended Levine’s result to the nonlinear case (\(q>1\)). They showed that solutions with a negative initial energy blow up in finite time. Later, Vitillaro [11] extended these results to the case of a nonlinear damping and a positive initial energy.
In [12], Messaoudi studied decay of solutions of the problem (1), using the techniques combination of the perturbed energy and potential well methods. Recently, the problem (1) was studied by Wu and Xue [13]. They proved the uniform energy decay rates of the solutions, by utilizing the multiplier method.
In this work, we established the polynomial and exponential decay of solutions of the problem (1) by using Nakao’s inequality. After that, we show the blow up of solutions with negative and nonnegative initial energy, using the same techniques as in [14].
This work is organized as follows: In the next section, we present some lemmas, notations, and a local existence theorem. In Section 3, the global existence and decay of solutions are given. In Section 4, we show the blow up of solutions, for \(q=1\).
2 Preliminaries
In this section, we shall give some assumptions and lemmas which will be used throughout this paper. Let \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{p}\) denote the usual \(L^{2} ( \Omega ) \) norm and \(L^{p} ( \Omega ) \) norm, respectively.
Lemma 1
(Sobolev-Poincaré inequality) [15]
Let p be a number with \(2\leq p<\infty\) (\(n=1,2\)) or \(2\leq p\leq\frac{2n}{n-2}\) (\(n\geq3\)), then there is a constant \(C_{\ast}=C_{\ast} ( \Omega , p ) \) such that
Lemma 2
[16]
Let \(\phi ( t ) \) be a nonincreasing and nonnegative function defined on \([ 0,T ] \), \(T>1\), satisfying
for \(w_{0}\) a positive constant and α a nonnegative constant. Then we have, for each \(t\in [ 0,T ] \),
where \([ t-1 ] ^{+}=\max \{ t-1,0 \} \) and \(w_{1}=\ln ( \frac{w_{0}}{w_{0}-1} ) \).
Next, we state the local existence theorem that can be established by combining the arguments of [6, 17, 18].
Theorem 3
(Local existence)
Suppose that \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\), and further \(u_{0}\in W_{0}^{1,m+2} ( \Omega ) \) and \(u_{1}\in L^{2} ( \Omega ) \) such that problem (1) has a unique local solution,
Moreover, at least one of the following statements holds true:
-
(i)
\(T=\infty\),
-
(ii)
\(\Vert u_{t}\Vert ^{2}+\Vert \nabla u\Vert _{m+2}^{m+2}\rightarrow\infty\) as \(t\rightarrow T^{-}\).
3 Global existence and decay of solutions
In this section, we discuss the global existence and decay of the solution for problem (1).
We define
and
We also define the energy function as follows:
Finally, we define
The next lemma shows that our energy functional (6) is a nonincreasing function along the solution of (1).
Lemma 4
\(E ( t ) \) is a nonincreasing function for \(t\geq0\) and
Proof
Multiplying the equation of (1) by \(u_{t}\) and integrating over Ω, using integrating by parts, we get
□
Lemma 5
Let \(u_{0}\in W\) and \(u_{1}\in L^{2} ( \Omega ) \). Suppose that \(p>m+1\) and
then \(u\in W\) for each \(t\geq0\).
Proof
Since \(I ( 0 ) >0\), it follows by the continuity of \(u ( t ) \) that
for some interval near \(t=0\). Let \(T_{m}>0\) be a maximal time, when (5) holds on \([ 0,T_{m} ] \).
Thus, from (6) and \(E ( t ) \) being nonincreasing by (8), we have
And so, exploiting Lemma 1, (10), and (12), we obtain
Therefore, by using (5), we conclude that \(I ( t ) >0\) for all \(t\in [ 0,T_{m} ] \). By repeating the procedure, \(T_{m}\) is extended to T. The proof of Lemma 5 is completed. □
Lemma 6
Let the assumptions of Lemma 5 hold. Then there exists \(\eta _{1}=1-\beta\) such that
Proof
From (13), we get
□
Let \(\eta_{1}=1-\beta\), then we have the following result.
Remark 7
From Lemma 6, we can deduce that
Theorem 8
Suppose that \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\) holds. Let \(u_{0}\in W\) satisfying (10). Then the solution of problem (1) is global.
Proof
It is sufficient to show that \(\Vert u_{t}\Vert ^{2}+ \Vert \nabla u\Vert _{m+2}^{m+2}\) is bounded independently of t. To achieve this we use (5) and (6) to obtain
since \(I ( t ) \geq0\). Therefore,
where \(C=\max \{ 2,\frac{ ( p+1 ) ( m+2 ) }{p-m-1} \} \). Then by Theorem 3, we have the global existence result. □
Theorem 9
Suppose that \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\), and (10) hold, and further \(u_{0}\in W\). Thus, we have the following decay estimates:
where \(w_{1}\), α, and \(C_{7}\) are positive constants which will be defined later.
Proof
By integrating (8) over \([ t,t+1 ] \), \(t>0\), we have
where
By virtue of (16) and Hölder’s inequality, we observe that
Hence, from (17), there exist \(t_{1}\in [ t,t+\frac {1}{4} ] \) and \(t_{2}\in [ t+\frac{3}{4},t+1 ] \) such that
Multiplying (1) by u and integrating it over \(\Omega\times [ t_{1},t_{2} ] \), using integration by parts, we get
By using (1) and integrating by parts and applying the Cauchy-Schwarz inequality in the first term and the Hölder inequality in the second term of the right-hand side of (19), we obtain
Now, our goal is to estimate the last term in the right-hand side of inequality (20). By using Hölder inequality, we obtain
By applying the Sobolev-Poincaré inequality and (12), we find
Now, we estimate the fourth term of the right-hand side of inequality (20). By using the embedding \(L^{m+2} ( \Omega ) \hookrightarrow L^{2} ( \Omega ) \), we have
which implies
Then
From (12), (18), and the Sobolev-Poincaré inequality, we have
where \(C_{1}=2C_{\ast} ( \frac{ ( p+1 ) ( m+2 ) }{p-m-1}E ( 0 ) ) ^{\frac{1}{m+2}}\). Then by (20)-(24) we have
On the other hand, from (5), (6), and Remark 7, we obtain
where \(C_{3}=\frac{1}{\eta_{1}}\frac{p-m-1}{ ( p+1 ) ( m+2 ) }+\frac{1}{p+1}\).
By integrating (26) over \([ t_{1},t_{2} ] \), we have
Then by (18), (25), and (27), we get
By integrating (8) over \([ t,t_{2} ] \), we obtain
Therefore, since \(t_{2}-t_{1}\geq\frac{1}{2}\), we conclude that
That is,
Consequently, exploiting (15), (28)-(30), and since \(t_{1},t_{2}\in [ t,t+1 ] \), we get
Then, by (28), we have
Hence, we obtain
Note that, since \(E ( t ) \) is nonincreasing and \(E ( t ) \geq0\) on \([ 0,\infty ) \),
Thus, we have
It follows from (32) and (33) that
Thus, we get
Case 1: When \(q=1\) and \(m=0\) from (34), we obtain
By Lemma 2, we get
where \(w_{1}=\ln\frac{C_{7}}{C_{7}-1}\).
Case 2: When \(( m+1 ) q>1\), applying Lemma 2 to (34) yield
where \(\alpha=\frac{ ( m+1 ) q-1}{m+2}\). The proof of Theorem 9 is completed. □
4 Blow up of solutions
In this section, we deal with the blow up of the solution for the problem (1), when \(q=1\). Let us begin by stating the following two lemmas, which will be used later.
Lemma 10
[14]
Let us have \(\delta>0\) and let \(B ( t ) \in C^{2} ( 0,\infty ) \) be a nonnegative function satisfying
If
with \(r_{2}=2 ( \delta+1 ) -2\sqrt{ ( \delta+1 ) \delta}\), then \(B^{\prime} ( t ) >K_{0}\) for \(t>0\), where \(K_{0}\) is a constant.
Lemma 11
[14]
If \(H ( t ) \) is a nonincreasing function on \([ t_{0},\infty ) \) and satisfies the differential inequality
where \(a>0\), \(b\in R\), then there exists a finite time \(T^{\ast}\) such that
Upper bounds for \(T^{\ast}\) are estimated as follows:
-
(i)
If \(b<0\) and \(H ( t_{0} ) <\min \{ 1,\sqrt{-\frac {a}{b}} \} \) then
$$ T^{\ast}\leq t_{0}+\frac{1}{\sqrt{-b}}\ln\frac{\sqrt{-\frac {a}{b}}}{\sqrt{-\frac{a}{b}}-H ( t_{0} ) }. $$ -
(ii)
If \(b=0\), then
$$ T^{\ast}\leq t_{0}+\frac{H ( t_{0} ) }{H^{\prime} ( t_{0} ) }. $$ -
(iii)
If \(b>0\), then
$$ T^{\ast}\leq\frac{H ( t_{0} ) }{\sqrt{a}}\quad \textit{or}\quad T^{\ast }\leq t_{0}+2^{\frac{3\delta+1}{2\delta}}\frac{\delta c}{\sqrt{a}} \bigl[ 1- \bigl( 1+cH ( t_{0} ) \bigr) ^{-\frac{1}{2\delta}} \bigr] , $$where \(c= ( \frac{a}{b} ) ^{2+\frac{1}{\delta}}\).
Definition 12
A solution u of (1) with \(q=1\) is called blow up if there exists a finite time \(T^{\ast}\) such that
Let
Lemma 13
Assume \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\), and that \(m\leq4\delta\leq p-1\), then we have
Proof
From (39), we have
Then from (6) and (42), we have
Since \(m\leq4\delta\leq p-1\), we obtain (40). □
Lemma 14
Assume \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\) and one of the following statements are satisfied:
-
(i)
\(E ( 0 ) <0\),
-
(ii)
\(E ( 0 ) =0\) and \(\int_{\Omega}u_{0}u_{1}\,dx>0\),
-
(iii)
\(E ( 0 ) >0\) and
$$ a^{\prime} ( 0 ) >r_{2} \biggl[ a ( 0 ) +\frac{K_{1}}{4 ( \delta+1 ) } \biggr] +\Vert u_{0}\Vert ^{2} $$(43)holds.
Then \(a^{\prime} ( t ) >\Vert u_{0}\Vert ^{2}\) for \(t>t^{\ast}\), where \(t_{0}=t^{\ast}\) is given by (44) in case (i) and \(t_{0}=0\) in cases (ii) and (iii).
Here \(K_{1}\) and \(t^{\ast}\) are defined in (48) and (44), respectively.
Proof
(i) If \(E ( 0 ) <0\), then from (40), we have
Thus we get \(a^{\prime} ( t ) >\Vert u_{0}\Vert ^{2}\) for \(t>t^{\ast}\), where
(ii) If \(E ( 0 ) =0\) and \(\int_{\Omega}u_{0}u_{1}\,dx>0\), then \(a^{\prime\prime} ( t ) \geq0\) for \(t\geq0\). We have \(a^{\prime} ( t ) >\Vert u_{0}\Vert ^{2}\), \(t\geq0\).
(iii) If \(E ( 0 ) >0\), we first note that
By the Hölder inequality and the Young inequality, we have
By the Hölder inequality, the Young inequality, and (46), we have
Hence, by (40) and (47), we obtain
where
Let
Then \(b ( t ) \) satisfies Lemma 10. Consequently, we get from (43) \(a^{\prime} ( t ) >\Vert u_{0}\Vert ^{2}\), \(t>0\), where \(r_{2}\) is given in Lemma 10. □
Theorem 15
Assume \(m+2< p+1<\frac{n ( m+2 ) }{n- ( m+2 ) }\), \(m+2< n\) and one of the following statements are satisfied:
-
(i)
\(E ( 0 ) <0\),
-
(ii)
\(E ( 0 ) =0\) and \(\int_{\Omega}u_{0}u_{1}\,dx>0\),
-
(iii)
\(0< E ( 0 ) <\frac{ ( a^{\prime} ( t_{0} ) -\Vert u_{0}\Vert ^{2} ) ^{2}}{8 [ a ( t_{0} ) + ( T_{1}-t_{0} ) \Vert u_{0}\Vert ^{2} ] }\) and (43) holds.
Then the solution u blow up in finite time \(T^{\ast}\) in the case of (38). In case (i),
Furthermore, if \(H ( t_{0} ) <\min \{ 1,\sqrt{-\frac {a}{b}} \} \), we have
where
In case (ii),
In case (iii),
where a and b are given; see (51), (52).
Proof
Let
where \(T_{1}>0\) is a certain constant which will be specified later. Then we get
and
where
For simplicity of the calculation, we define
From (41), (45), and the Hölder inequality, we get
If case (i) or (ii) holds, by (40) we have
Thus, from (59)-(61) and (55), we obtain
From (39),
and (55), we get
where
By the Schwarz inequality, and \(\Theta ( t ) \) being nonnegative, we have
Therefore, by (58) and (62), we get
By Lemma 13, we know that \(H^{\prime} ( t ) <0\) for \(t\geq t_{0}\). Multiplying (63) by \(H^{\prime} ( t ) \) and integrating it from \(t_{0}\) to t, we get
for \(t\geq t_{0}\), where a, b are defined in (51) and (52) respectively.
If case (iii) holds, by the steps of case (i), we get \(a>0\) if and only if
Then by Lemma 11, there exists a finite time \(T^{\ast}\) such that \(\lim_{t\rightarrow T^{\ast-}}H ( t ) =0\) and the upper bound of \(T^{\ast}\) is estimated according to the sign of \(E ( 0 )\). This means that (38) holds. □
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Pişkin, E. On the decay and blow up of solutions for a quasilinear hyperbolic equations with nonlinear damping and source terms. Bound Value Probl 2015, 127 (2015). https://doi.org/10.1186/s13661-015-0395-4
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DOI: https://doi.org/10.1186/s13661-015-0395-4