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

From the introduction: Suppose Ω 3 is a bounded smooth domain, gL 2 (Ω). We consider the following equation

u t -Δu t -Δu=f(u)+g(x)inΩ,u=0onΩ·

We impose on the nonlinear function f the following dissipative condition

lim sup |s| f(s) s<λ 1 ,

where λ 1 is the first eigenvalue of A=-Δ with domain D(A)=H 2 (Ω)H 0 1 (Ω), and the growth conditions |f ' (x)|c(1+|s| 4 ), s, and |f(s)|c(1+|s| γ ), γ<5, s.

Under the above hypotheses, equations (1) and (2) will define a C 0 -semigroup S(t) on the Hilbert space V=H 0 1 (Ω). Our main result is that there exist a global attractor.

35B41Attractors (PDE)
35K35Higher order parabolic equations, boundary value problems
35K70Ultraparabolic equations, pseudoparabolic equations, etc.
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