## Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation.(English)Zbl 0666.35012

For a smooth bounded domain $$\Omega \subset R^ n$$, $$n\leq 3$$, and $$\epsilon$$ a real parameter, consider the hyperbolic equation $\epsilon u_{tt}+u_ t-\Delta u=-f(u)-g\quad in\quad \Omega$ with Dirichlet boundary conditions. Under certain conditions on the function f(u), this equation has a compact attractor $${\mathcal A}_{\epsilon}$$ in $$H^ 1_ 0\times L^ 2$$. For $$\epsilon =0$$, the parabolic equation also has a compact attractor which can be naturally embedded into a compact set $${\mathcal A}_ 0$$ in $$H^ 1_ 0\times L_ 2$$. The authors demonstrate that, for any neighborhood of U, the set $${\mathcal A}_{\epsilon}\subset U$$ for $$\epsilon$$ small.
Reviewer: M.Witten

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 35L70 Second-order nonlinear hyperbolic equations
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### References:

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