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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 ${ℝ}^{n}$ $\left(n\ge 3\right)$ of the form

${u}_{t}={\Delta }u-\lambda u-f\left(u\right)+{f}_{{x}_{i}}^{i}+g\left(x\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{+}×{ℝ}^{n}\phantom{\rule{2.em}{0ex}}\left(*\right)$

with initial data

$u\left(0,x\right)={u}_{0}\left(x\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{n}\phantom{\rule{2.em}{0ex}}\left(**\right)$

is shown. Here, the nonlinearity $f$ is allowed to have polynomial growth of arbitrary order $p-1$ $\left(p\ge 2\right)$ and in the inhomogeneous term, ${f}_{{x}_{i}}^{i}$, $i=1,...,n$, are distributional derivatives of $f\in {L}^{2}\left({ℝ}^{n}\right)$, $g\in {L}^{2}\left({ℝ}^{n}\right)$.

In order to obtain this for the problem $\left(*\right)$, $\left(**\right)$ 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 ${L}^{p}\left({ℝ}^{n}\right)$-topology without restriction on $p$.

Thus, for abstract semigroups in ${L}^{2}\left({ℝ}^{n}\right)$ the authors derive a sufficient criterion that a global attractor in ${L}^{2}\left({ℝ}^{n}\right)$ also attracts bounded sets of ${L}^{2}\left({ℝ}^{n}\right)$ w.r.t. the ${L}^{p}\left({ℝ}^{n}\right)$-norm.

Using a new method based on a priori estimates, this criterion applies to show that the semigroup in ${L}^{2}\left({ℝ}^{2}\right)$ associated with $\left(*\right)$, $\left(**\right)$ possesses a $\left({L}^{2}\left({ℝ}^{n}\right),{L}^{p}\left({ℝ}^{n}\right)\right)$-global attractor $A$ in the sense that $A$ is nonempty, compact, invariant in ${L}^{p}\left({ℝ}^{n}\right)$ and attracts every bounded subset of ${L}^{2}\left({ℝ}^{n}\right)$ in the ${L}^{p}\left({ℝ}^{n}\right)$-norm.

MSC:
 35B41 Attractors (PDE) 35K57 Reaction-diffusion equations 35B45 A priori estimates for solutions of PDE 35K15 Second order parabolic equations, initial value problems