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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: $u_{tt}+Bu_t+Au=f(x,u),\qquad u(0,\cdot)=u_0(\cdot),\quad u_t(0,\cdot)=u_1(\cdot).$

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