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Attractors for a nonclassical diffusion equation. (English) Zbl 1017.35025

From the introduction: Suppose \(\Omega\subset\mathbb{R}^3\) is a bounded smooth domain, \(g\in L^2(\Omega)\). We consider the following equation \[ \begin{aligned} u_t-\Delta u_t-\Delta u=f(u)+g(x) \quad & \text{in }\Omega, \tag{1}\\ u=0 \quad & \text{on }\partial \Omega.\tag{2}\end{aligned} \] We impose on the nonlinear function \(f\) the following dissipative condition \[ \limsup_{|s|\to \infty} {f(s) \over s}< \lambda_1, \] where \(\lambda_1\) is the first eigenvalue of \(A=-\Delta\) with domain \(D(A)=H^2 (\Omega)\cap H^1_0(\Omega)\), and the growth conditions \(|f'(x)|\leq c(1+|s|^4)\), \(s\in\mathbb{R}\), and \(|f(s)|\leq c(1+ |s|^\gamma)\), \(\gamma<5\), \(s\in\mathbb{R}\).
Under the above hypotheses, equations (1) and (2) will define a \(C^0\)-semigroup \(S(t)\) on the Hilbert space \(V=H^1_0(\Omega)\). Our main result is that there exist a global attractor.

MSC:

35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
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