On the Wave Equation with Hyperbolic Dynamical Boundary Conditions, Interior and Boundary Damping and Source

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• 作者：Enzo Vitillaro
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：223
• 期：3
• 页码：1183-1237
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0673
• 卷排序：223

文摘

The aim of this paper is to study the problem$$\left\{\begin{array}{ll} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \quad & {\rm in} \, (0,\infty)\times\Omega, \\ u=0 & {\rm on} \, (0,\infty)\times \Gamma_0, \\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t)=g(x,u)\quad & {\rm on} \, (0,\infty)\times \Gamma_1,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & {\rm in} \, \overline \Omega, \end{array}\right.$$ where \({\Omega}\) is a open bounded subset of \({{\mathbb R}^N}\) with C1 boundary (\({N \ge 2}\)), \({\Gamma = \partial\Omega}\), \({(\Gamma_{0},\Gamma_{1})}\) is a measurable partition of \({\Gamma}\), \({\Delta_{\Gamma}}\) denotes the Laplace–Beltrami operator on \({\Gamma}\), \({\nu}\) is the outward normal to \({\Omega}\), and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear subcritical perturbations. In the paper a local Hadamard well-posedness result for initial data in the natural energy space associated to the problem is given. Moreover, when \({\Omega}\) is C2 and \({\overline{\Gamma_{0}} \cap \overline{\Gamma_{1}} = \emptyset}\), the regularity of solutions is studied. Next a blow-up theorem is given when P and Q are linear and f and g are superlinear sources. Finally a dynamical system is generated when the source parts of f and g are at most linear at infinity, or they are dominated by the damping terms.Communicated by A. Bressan