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Exponential attractors for reaction-diffusion equations with arbitrary polynomial growth. (English) Zbl 1170.35349
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 ${L}^{2}\left(𝛺\right)$ with $g\in {H}^{-1}\left(𝛺\right)$. Second, by proving the semigroup satisfying discrete squeezing property, we obtain the exponential attractor in ${H}_{0}^{1}\left(𝛺\right)$ with $g\in {L}^{2}\left(𝛺\right)$. Because the solutions without higher regularity than ${L}^{2p-2}\left(𝛺\right)$ for $g$ belong only to ${L}^{2}\left(𝛺\right)$ in the equation, the general method by proving the Lipschitz continuity between ${L}^{2p-2}\left(𝛺\right)$ and ${L}^{2}\left(𝛺\right)$ 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., ${L}^{2p-2}\left(𝛺\right)$ with $g\in 𝔾$ (stated in a definition) when it is out of reach for the other known techniques.
##### MSC:
 35B41 Attractors (PDE) 35K57 Reaction-diffusion equations