Some nonlinear wave equations with acoustic boundary conditions. (English) Zbl 0979.35105

Lopes Frota, Cícero; Goldstein, Jerome A.

J. Differ. Equations 164, No. 1, 92-109 (2000).

The authors consider a boundary value problem for the Carrier equation with nonlinear dissipation and acoustic boundary conditions: \[ u_{tt}-M(\|u\|^2_{L^2})\Delta u+\beta|u_t|^\alpha u_t=0\text{ in }\Omega\times (0,T),\tag{1} \] where \(\Omega\) is a bounded region of \(\mathbb{R}^n\) with smooth boundary \(\Gamma=\Gamma_0\cup\Gamma_1\). Here \(\Gamma_0,\Gamma_1\) have nonempty interior. The boundary conditions for (1) are \[ \begin{cases} u=0\text{ on } \Gamma_0\times (0,T),\\ \rho u_t+f w_{tt}+gw_t+hw=0\text{ on }\Gamma_1\times (0,T),\\ \frac{\partial u}{\partial \eta}=w_t\text{ on }\Gamma_1\times (0,T),\end{cases}\tag{2} \] where \(f(x)\), \(g(x)\), and \(h(x)\) are continuous real functions on \(\overline{\Gamma_1}\) satisfying \(f(x)> 0\), \(h(x) > 0\), and \(g(x) > 0\) for all \(x\in \overline\Gamma_1\). In (1) and (2), \(\alpha >1\), \(\beta\) and \(\rho\) are positive constants, and the function \(M\in C^1([0,+\infty);\mathbb{R})\), satisfies \(0 < m_0\leq M(s)\) as well as \(|M'(s)\sqrt{s}|\leq m_1M(s)\) for all \(s\geq 0\) and some positive constants \(m_0\) and \(m_1\). The authors prove global existence of a pair of functions \(\{u(x,t) , w(x,t)\}\) satisfying (1) and (2) and belonging to the space \(V\cap Y\times L^2(\Gamma)\) provided \(\alpha> 1\), and the initial data belong to \(V\cap H^2(\Omega)\times V\cap L^{2\alpha+2}(\Omega),\) where \(Y=\{v\in H^1(\Omega),\;\Delta v\in L^2(\Omega)\}\) and \(V=\{u\in H^1(\Omega),\;\delta_0u=0\) a.e. on \(\Gamma_0\}.\) Furthermore, if the dimension \(n\) is either 1 or 2 and \(\alpha > 1\) or \(n = 3\) and \(1 < \alpha \leq 2\) then, they prove uniqueness, regularity results and continuous dependence on the parameters. The results are proved using the Galerkin procedure and the energy method. The article is clearly written.

Reviewer: Gustavo A.Perla Menzala (Petropolis))

MSC:

35L70	Second-order nonlinear hyperbolic equations
35L20	Initial-boundary value problems for second-order hyperbolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Carrier equation; nonlinear dissipation; Galerkin procedure

References:

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