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Attractors for semilinear damped wave equations on \(\mathbb{R}^ 3\). (English) Zbl 0819.35096

The paper deals with the long-time behavior of solutions of the problem: \[ u_{tt}+ du_ t- \Delta u+ f(x,u) =0,\;x\in \mathbb{R}^ 3,\;t\geq 0,\;u(.,0)= u^ 0,\;u_ t (.,0)= u^ 1\;(d>0), \] where \((u^ 0, u^ 1)\in H^ 1 (\mathbb{R}^ 3) \times L^ 2 (\mathbb{R}^ 3)\). Assume: \(f\in C^ 2 (\mathbb{R}^ 3)\), \(f(.,0)\in H^ 1 (\mathbb{R}^ 3)\), \(|\partial f(x,0)/ \partial z|\leq c\) for all \(x\in \mathbb{R}^ 3\); \(| \partial^ 2 f(x,z)/ \partial z^ 2|\leq c(1+| z|)\) for \(x\in \mathbb{R}^ 3\), \(z\in \mathbb{R}^ 1\); \(\liminf_{| z|\to \infty} f(x,z)/ z\geq 0\) uniformly in \(x\in \mathbb{R}^ 3\); \((f(x,z)- f(x,0))z\geq cz^ 2\), \(c>0\) for all \(z\in \mathbb{R}^ 1\), \(| x|\geq r_ 1\). The author proves that under these hypotheses, there exists a (unique) global attractor of the semigroup \(\{S_ t\}\), \(S_ t (u^ 0, u^ 1)= (u(t), u_ t(t))\), \(t\geq 0\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
47H20 Semigroups of nonlinear operators
35B40 Asymptotic behavior of solutions to PDEs
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