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Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains. (English) Zbl 1082.35036
In this well-written paper, the existence of a global attractor for nonlinear reaction-diffusion equations in ${\Bbb R}^n$ $(n\geq 3)$ of the form $$ u_t=\Delta u-\lambda u-f(u)+f_{x_i}^i+g(x) \quad\text{in }{\Bbb R}^+\times{\Bbb R}^n \tag*$$ with initial data $$u(0,x)=u_0(x) \quad\text{in }{\Bbb R}^n \tag**$$ is shown. Here, the nonlinearity $f$ is allowed to have polynomial growth of arbitrary order $p-1$ $(p\geq 2)$ and in the inhomogeneous term, $f_{x_i}^i$, $i=1,\ldots,n$, are distributional derivatives of $f\in L^2({\Bbb R}^n)$, $g\in L^2({\Bbb R}^n)$. In order to obtain this for the problem $(\ast)$, $(\ast\ast)$ two difficulties appear: (1) The regularity of its solutions is not sufficiently high to apply appropriate embedding theorems. (2) It is hard to get continuity of the associated semigroup in the $L^p({\Bbb R}^n)$-topology without restriction on $p$. Thus, for abstract semigroups in $L^2({\Bbb R}^n)$ the authors derive a sufficient criterion that a global attractor in $L^2({\Bbb R}^n)$ also attracts bounded sets of $L^2({\Bbb R}^n)$ w.r.t. the $L^p({\Bbb R}^n)$-norm. Using a new method based on a priori estimates, this criterion applies to show that the semigroup in $L^2({\Bbb R}^2)$ associated with $(\ast)$, $(\ast\ast)$ possesses a $(L^2({\Bbb R}^n),L^p({\Bbb R}^n))$-global attractor $A$ in the sense that $A$ is nonempty, compact, invariant in $L^p({\Bbb R}^n)$ and attracts every bounded subset of $L^2({\Bbb R}^n)$ in the $L^p({\Bbb R}^n)$-norm.

MSC:
35B41Attractors (PDE)
35K57Reaction-diffusion equations
35B45A priori estimates for solutions of PDE
35K15Second order parabolic equations, initial value problems
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References:
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