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On a nonlinear wave equation with boundary damping. (English) Zbl 1292.35049
Summary: This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation \[ u''-\mu\Delta u+\alpha f\biggl(\int_\Omega u^2dx\biggl)u+\beta g\biggl(\int_\Omega u^{\prime^2}dx\biggl)u'=0\quad\text{in}\quad\Omega\times(0,\infty) \] with boundary conditions \(u=0\text{ on }\Gamma_0\times(0,\infty)\text{ and }\frac{\partial u}{\partial\nu}+h(\cdot,u')=0\text{ on }\Gamma_1\times (0,\infty)\). Here, \(\Omega \) is an open bounded set of \(\mathbb R^n\) with boundary \(\Gamma\) of class \(C^{2}\); \(\Gamma \) is constituted of two disjoint closed parts \(\Gamma _{0}\) and \(\Gamma _{1}\) both with positive measure; the functions \(\mu (t)\), \(f(s)\), \(g(s)\) satisfy the conditions \(\mu (t)\geq \mu _{0}>0\), \(f(s)\geq 0\), \(g(s)\geq 0\) for \(t\geq 0\), \(s\geq 0\) and \(h(x,s)\) is a real function where \(x\in\Gamma_{1}\), \(s\in\mathbb R\); \(\nu (x)\) is the unit outward normal vector at \(x\in\Gamma_1\) and \(\alpha\), \(\beta \) are non-negative real constants.
Assuming that \(h(x,s)\) is strongly monotone in \(s\) for each \(x\in \Gamma _{1}\), it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao’s method.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35R09 Integral partial differential equations
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