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Zbl 1170.35349
Zhong, Yansheng; Zhong, Chengkui
Exponential attractors for reaction-diffusion equations with arbitrary polynomial growth. (English)
[J] Nonlinear Anal., Theory Methods Appl. 71, No. 3-4, A, 751-765 (2009). ISSN 0362-546X

Summary: We study exponential attractors for an equation with arbitrary polynomial growth nonlinearity $f$ and inhomogeneous term $g$. First, we prove by the $\ell $-trajectory method that the exponential attractor in $L^{2}(\varOmega )$ with $g\in H^{ - 1}(\varOmega )$. Second, by proving the semigroup satisfying discrete squeezing property, we obtain the exponential attractor in $H_0^1(\varOmega)$ with $g\in L^{2}(\varOmega )$. Because the solutions without higher regularity than $L^{2p - 2}(\varOmega )$ for $g$ belong only to $L^{2}(\varOmega )$ in the equation, the general method by proving the Lipschitz continuity between $L^{2p - 2}(\varOmega )$ and $L^{2}(\varOmega )$ does not work in our case. Therefore, we give a new method (presented in a theorem) to obtain an exponential attractor in a stronger topology space i.e., $L^{2p - 2}(\varOmega )$ with $g\in \Bbb G$ (stated in a definition) when it is out of reach for the other known techniques.

MSC 2000:: *35B41 Attractors
35K57 Reaction-diffusion equations

Keywords: exponential attractor; asymptotic a priori estimate; discrete squeezing property; semigroup; supercritical nonlinearity; global attractor; polynomial growth nonlinearity

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Zentralblatt MATH Berlin [Germany]