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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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