Existence and continuity of the attractor for a singularly perturbed hyperbolic equation

Abstract

In a bounded smooth domain domain Ω⊂ℝⁿ the nonlinear hyperbolic evolutionary equation depending on a small parameter ε>0

$$\varepsilon \partial _t^2 u + \gamma \partial _t u + b(\partial _t u) = \Delta u - f(u) + g(x),u|_{\partial \Omega } = 0$$

((1))

$$u|_{t = 0} = u_0 ,\partial _t u|_{t = 0} = u_1 $$

((2))

whereu(x, t) is a real-valued function ofx ∃ Ω, t ∃ ℝ₊, is considered. In the case of monotone damping, i.e.,γ>0 andb is a nondecreasing function, and further suitable conditions supposed, the existence of a global compact attractorA(ε) for the semigroup generated in the spaceH ¹₀ (Ω)×L ²(Ω) by (1)–(2) is shown. In the case of singular perturbation, the problem of upper and lower semicontinuity of the attractor asε→0⁺ is investigated.