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Zbl 1017.35025
Xiao, Yuelong
Attractors for a nonclassical diffusion equation. (English)
[J] Acta Math. Appl. Sin., Engl. Ser. 18, No.2, 273-276 (2002). ISSN 0168-9673; ISSN 1618-3932/e

From the introduction: Suppose $\Omega\subset\bbfR^3$ is a bounded smooth domain, $g\in L^2(\Omega)$. We consider the following equation $$\align u_t-\Delta u_t-\Delta u=f(u)+g(x) \quad & \text{in }\Omega, \tag 1\\ u=0 \quad & \text{on }\partial \Omega.\tag 2\endalign$$ 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 $|f'(x)|\le c(1+|s|^4)$, $s\in\bbfR$, and $|f(s)|\le c(1+ |s|^\gamma)$, $\gamma<5$, $s\in\bbfR$.\par 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 2000:: *35B41 Attractors
35K35 Higher order parabolic equations, boundary value problems
35K70 Ultraparabolic (etc.) problems

Keywords: dissipative conditions; growth conditions; asymptotic behavior; existence of a global attractor

Cited in: Zbl 1176.35178

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