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Almost-periodic forcing for a wave equation with a nonlinear local damping term. (English) Zbl 0589.35076
Summary: Let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded open domain and \(\Gamma =\partial \Omega\). If \(\beta\) is a maximal monotone graph in \({\mathbb{R}}\times {\mathbb{R}}\) with \(0\in \beta (0)\), \(f: {\mathbb{R}}\times \Omega \to {\mathbb{R}}\) is measurable with \(t\to f(t,.)\) \(S^ 2\)-almost periodic as a function \({\mathbb{R}}\to L^ 2(\Omega)\), we consider the nonlinear hyperbolic equation \[ (1)\quad \partial^ 2u/\partial t^ 2-\Delta u+\beta (\partial u/\partial t)\ni f(t,x)\quad on\quad {\mathbb{R}}^+\times \Omega,\quad u(t,x)=0,\quad on\quad {\mathbb{R}}^+\times \Gamma. \] We show that: (i) If \(\beta\) is strictly increasing and (1) has a solution \(\omega\) on \({\mathbb{R}}\) with [\(\omega\),\(\partial \omega /\partial t]\) almost periodic: \({\mathbb{R}}\to H^ 1_ 0(\Omega)\times L^ 2(\Omega)\), for any solution of (1) there exists \(\xi (x)\in H^ 1_ 0(\Omega)\) with u(t,.)-\(\omega\) (t,.)\(\rightharpoonup \xi\) in \(H^ 1_ 0(\Omega)\) as \(t\to +\infty;\)
(ii) if \(\beta\) is single valued and everywhere defined, the existence of \(\omega\) as above implies that, for every solution of (1), there exists \(\zeta\) (t,x) with \(\partial^ 2\zeta /\partial t^ 2-\Delta \zeta =0\) in \({\mathbb{R}}\times \Omega\) and \(u(t,.)-\omega (t,.)-\xi (t,.)\rightharpoonup 0\) in \(H^ 1_ 0(\Omega)\) as \(t\to +\infty;\)
(iii) if \(\beta^{-1}\) is uniformly continuous and \(\beta\) satisfies some growth assumption (depending on N), for every f as above, there exists \(\omega\) solution of (1) on \({\mathbb{R}}\) with [\(\omega\),\(\partial \omega /\partial t]\) almost periodic: \({\mathbb{R}}\to H^ 1_ 0(\Omega)\times L^ 2(\Omega)\).

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B15 Almost and pseudo-almost periodic solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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