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Global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions
Boundary Value Problems volume 2017, Article number: 42 (2017)
Abstract
We consider the quasilinear wave equation
\(a,b>0\), associated with initial and Dirichlet boundary conditions at one part and acoustic boundary conditions at another part, respectively. We prove, under suitable conditions on α, β, m, p and for negative initial energy, a global nonexistence of solutions.
1 Introduction
In this paper, we consider the following quasilinear wave equation with acoustic boundary conditions:
where \(a,b>0, \alpha,\beta,m,p >2\), Ω is a regular and bounded domain of \(R^{n}(n \geq1)\) and \(\partial\Omega(=\Gamma):= \Gamma_{0} \cup \Gamma_{1}\). Here \(\Gamma_{0}, \Gamma_{1}\) are closed and disjoint, and \(\frac {\partial}{\partial\nu}\) denotes the unit outer normal derivative. The functions \(f,q,h : \Gamma _{1} \longrightarrow R^{+}\) are essentially bounded and \(0< q_{0} \leq q(x)\) on \(\Gamma_{1}\).
The system (1.1)-(1.6) is a model of a quasilinear wave equation with acoustic boundary conditions. The acoustic boundary conditions were introduced by Morse and Ingard [1] in 1968 and developed by Beale and Rosencrans in [2], where the authors proved the global existence and regularity of the linear problem. Furthermore, Boukhatem and Benabderrahmane [3, 4] studied the existence, blow-up and decay of solutions for viscoelastic wave equations with acoustic boundary conditions. Graber and Said-Houari [5] studied the blow-up solutions for the wave equation with semilinear porous acoustic boundary conditions. Moreover, Wu [6] also considered blow-up solutions for a nonlinear wave equation with porous acoustic boundary conditions. The global nonexistence of solutions for a class of wave equations with nonlinear damping and source terms was proved by Messaoudi and Said-Houari [7–9] (see [10–13] for more details). Recently, Piskin [14] investigated the energy decay and blow-up of solutions for quasilinear hyperbolic equations with nonlinear damping and source terms (see [15–18] for more details).
Motivated by the previous works, in this paper, we study the global nonexistence of solutions for quasilinear wave equations with acoustic boundary conditions. To the best of our knowledge, there are no results of a quasilinear wave equation with acoustic boundary conditions. This work is meaningful. The outline of the paper is the following. In Section 2, we prove the main result.
2 Blow-up results
In order to state and prove our result, we introduce
for \(T>0\) and the energy functional
Theorem 2.1
Assume that \(\alpha, \beta, m, p \geq2\) such that \(\beta<\alpha\), and \(max\{m,\alpha\} < p< r_{\alpha}\), where \(r_{\alpha}\) is the Sobolev critical exponent of \(W^{1, \alpha}(\Omega)\). Assume further that
Then the solution \((u,y) \in Z \times L^{2}(R^{+}; L^{2}(\Gamma_{1}))\) of (1.1)-(1.6) can not exist for all time.
Remark 2.2
If the solution u of (1.1)-(1.6) is smooth enough, then it blows up in finite time.
Proof
We suppose that the solution exists for all time, and we reach a contradiction. For this purpose, we multiply Eq. (1.1) by \(u_{t}\) and, using (1.2)-(1.4), we obtain
for any regular solution. Hence we get \(E(t) \leq E(0)\) \(\forall t \geq0\).
By setting \(H(t)=-E(t)\), we deduce
Now, we define
for ε small to be chosen later and
Our goal is to show that \(L(t)\) satisfies a differential inequality of the form
This, of course, will lead to a blow-up in finite time.
By taking a derivative of (2.5), we get
By using Eqs. (1.1)-(1.4), estimate (2.8) becomes
Exploiting Hölder’s and Young’s inequalities, for any \(\eta,\mu ,\delta>0\), we obtain
A substitution of (2.10)-(2.12) in (2.9) yields
Therefore, by choosing \(\eta,\mu,\delta\) so that
for \(M_{1}, M_{2}, M_{3}\) to be specified later, and using (2.13), we arrive at
If \(M = M_{2}+ \frac{(\beta-1)M_{3}}{\beta}+\frac{(m-1)M_{1}}{m}\), then (2.14) takes the form
Then we use the embedding \(L^{p}(\Omega) \hookrightarrow L^{m}(\Omega)\) and (2.4) to get
We also exploit the inequality
the embedding \(W^{1,\alpha}(\Omega) \hookrightarrow H^{1} (\Omega)\) and (2.4) to obtain
Since \(\alpha> \beta\), we obtain
we derive
where c is a constant depending on Ω only. By using (2.6) and the inequality
we get the following inequalities:
and
where \(d=1+1/H(0), a=H(0)\). Inserting (2.16)-(2.18) and (2.20)-(2.22) into (2.15), we deduce
for some constant k and \(c_{1}= \frac{acd}{m} (\frac{b}{p})^{\sigma (m-1)}\), \(c_{2}= \frac{cd}{4} (\frac{b}{p})^{\sigma}\), \(c_{3}= \frac {cd}{\beta} (\frac{b}{p})^{\sigma(\beta-1)}\).
Using \(k=\varepsilon p\), we arrive at
At this point, by choosing \(M_{1}, M_{2}, M_{3}\) large enough and using
we have
where r is a positive constant (this is possible since \(p>\alpha\)).
We choose \(0<\varepsilon< \frac{1-\sigma}{M}\) so that
Then from (2.23) we get
and
On the other hand, from (2.5) and \(f, h >0\), we have
Consequently, the above estimate leads to
From Hölder’s inequality, we obtain
where c is the positive constant which comes from the embedding \(L^{\alpha}(\Omega) \hookrightarrow L^{2}(\Omega)\). This inequality implies that there exists a positive constant \(c_{4}>0\) such that
Applying Young’s inequality to the right-hand side of the preceding inequality, we have a positive constant, also denoted by \(c>0\), such that
for \(\frac{1}{\mu} + \frac{1}{\theta}=1\). We take \(\theta=2(1-\sigma)\), hence \(\mu=2(1-\sigma)/(1-2\sigma)\), to get
By Poincare’s inequality, we obtain
We use (2.6) and the algebraic inequality (2.19) with \(z=\Vert \nabla u(t) \Vert _{\alpha}^{\alpha}\), \(d= 1+ 1/H(0)\), \(a=H(0)\), \(\nu =2/{\alpha(1-2\sigma)}\), condition (2.6) on σ ensures that \(0<\nu<1\), and it follows that
Therefore, from (2.20), there exists a positive constant, denoted by \(c_{4}\), such that for all \(t \geq0\),
Furthermore, by the same method, we have
Using the embedding \(W_{0}^{1,\alpha}(\Omega) \hookrightarrow L^{2} (\Gamma _{1})\) and Hölder’s inequality, we get
Consequently, there exists a positive constant \(c_{5}=c_{5}( \Vert h\Vert _{\infty}, \Vert q\Vert _{\infty}, q_{0}, \sigma, \alpha)\) such that
Using Young’s inequality exactly as in (2.26), we write
where \(c_{6}\) is a positive constant depending on \(c_{5}\) and α. Consequently, applying once again the algebraic inequality (2.19) with \(z=\Vert \nabla u(t)\Vert _{\alpha}^{\alpha}\), \(\nu=2/{\alpha (1-2\sigma)}\) and making use of (2.6), we obtain by the same method as above
where \(c_{7}\) is a positive constant. From (2.25), (2.26) and (2.27), we arrive at
where c is a positive constant. Consequently, a combination of (2.24) and (2.28), for some \(\xi>0\), yields
Integration of (2.29) over \((0,t)\) gives
Hence \(L(t)\) blows up in finite time
Thus the proof of Theorem 2.1 is complete. □
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Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1A1B03930361).
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Jeong, JM., Park, JY. & Kang, Y.H. Global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions. Bound Value Probl 2017, 42 (2017). https://doi.org/10.1186/s13661-017-0773-1
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DOI: https://doi.org/10.1186/s13661-017-0773-1