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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.


MSC:
35B41Attractors (PDE)
35K35Higher order parabolic equations, boundary value problems
35K70Ultraparabolic equations, pseudoparabolic equations, etc.
References:
[1]Aifantis, E.C.: On the problem of diffusion in solids. Acta Mech., 37: 265–296 (1980) · Zbl 0447.73002 · doi:10.1007/BF01202949
[2]Hale, J.K.: Asymptotic behavior of dissipative systems. Math. Surv. Mono. 25, AMS Providence, 1980
[3]Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Springer-Verlag, Berlin, 1972
[4]Liu, Y.C., Wang, F.: A class of multidimensional nonlinear Sobolev-Galpern equation. Acta Math. Appl. Sinica, 17: 569–577 (1994) (in Chinese)
[5]Peter, J.C., Gurtin, M.E.: On the theory of heat conduction involving two temperatures. Z. Ange. Math. Phys., 19: 614–627 (1968) · Zbl 0159.15103 · doi:10.1007/BF01594969
[6]Temam, R.: Infinite dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, 1988
[7]Ting, T.W.: Certain non-steady flows of second order fluids. Arch. Rational Mech. Anal., 1963, 14: 1–26 · Zbl 0139.20105 · doi:10.1007/BF00250690
[8]Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. Encyclopedia of physics 3, Springer-Verlag, Berlin, 1965