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Attractors in ${L}^{2}\left({ℝ}^{N}\right)$ for a class of reaction-diffusion equations. (English) Zbl 1221.35203

The authors study the class of nonlinear reaction-diffusion equations:

$\begin{array}{cc}& \frac{\partial u}{\partial t}=b{\Delta }u-cu-f\left(u\right)-a\left(x\right)h\left(u\right)+g\left(x\right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},\hfill \\ & u\left(x,0\right)={u}_{0}\left(x\right)·\hfill \end{array}$

They prove the existence of a global attractor $A$ in ${L}^{2}\left({ℝ}^{N}\right)$ for the semigroup associated to this problem; $A$ is compact, invariant and attracts every bounded subsets of ${L}^{2}\left({ℝ}^{N}\right)$.

The proof relies on results of Q.-F. Ma, S.-H. Wang and C.-K. Zhong [Indiana Univ. Math. J. 51, No.,6, 1542–1558 (2002; Zbl 1028.37047)].

##### MSC:
 35K57 Reaction-diffusion equations 35B41 Attractors (PDE) 47H20 Semigroups of nonlinear operators 34C20 Transformation and reduction of ODE and systems, normal forms 34D20 Stability of ODE