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Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. (English) Zbl 0912.35028
In the paper is studied the asymptotic behaviour, as \(t\to \infty\), of global bounded solutions of the second order semilinear evolution problem \[ \begin{aligned} &u_{tt}+\alpha u_t=\Delta u+f(x,u)\quad \text{in } \mathbb{R}^+\times \Omega\\ &u=0\quad \text{on } \mathbb{R}^+\times \partial \Omega, \tag{1}\\ &u(0,\cdot)=u(\cdot), \quad u_t(0,\cdot)=u(\cdot)\quad \text{in } \Omega,\end{aligned} \] where \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^N, \alpha >0\) and \(f:\Omega \times \mathbb{R}\to \mathbb{R}\) is a smooth function. Assume that \(f(x,s)\) is analytic in \(s\), \(f(x,s), \frac{\partial f}{\partial s}(x,s), \frac{\partial ^2f}{\partial s^2}(x,s)\) are bounded in \(\Omega\times (-\beta,\beta)\), for all \(\beta>0\). Under this conditions the author proves that there exists \(p\geq 2\) such that \(\bigcup_{t\geq 0} \{u(t,\cdot),u(t,\cdot)\}\) is precompact in \(W^{2,p}(\Omega)\times W^{1,p}(\Omega)\), with \(p>N/2\) if \(N\leq 6\), and \(p>N\) if \(N>6\). Moreover, there exists \(\phi\in H^2(\Omega) \cap H^1_0(\Omega)\) with \(-\Delta \phi=f(x,\phi)\) in \(\Omega\) such that \[ \lim_{t\to +\infty} (\| u_t(t,\cdot)\|_{L^2(\Omega)}+ \| u(t,\cdot)-\phi(\cdot)\|_{W^{2,p}(\Omega)})=0. \] Let \(V\subset H=L^2(\Omega,\mathbb{R}^d)\) be a real Hilbert space, such that \(V\) is dense in \(H\) and the imbedding of \(V\) in \(H\) is compact. Let \(A\) be a linear unbounded self-adjoint operator associated with a bilinear continuous, symmetric and coercive form \(a(u,v)\) on \(V\), and \(B\) is a linear bounded operator \(B:H\to H\) such that \((Bx,x)_H\geq \alpha\| x\|_H\), \(\alpha>0\). Under suitable conditions on operators \(A, B\) and function \(f, f:A^{-1}(L^p(\Omega, \mathbb{R}^d))\to L^p(\Omega,\mathbb{R}^d)\) the author generalizes the result obtained for the problem (1) for the following problem: \[ u_{tt}+Bu_t+Au=f(x,u),\qquad u(0,\cdot)=u_0(\cdot),\quad u_t(0,\cdot)=u_1(\cdot). \]

35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI
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