Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1082.35036
Sun, Chunyou; Zhong, Chengkui
Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains. (English)
[J] Nonlinear Anal., Theory Methods Appl. 63, No. 1, A, 49-65 (2005). ISSN 0362-546X

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 $f\in L^2({\Bbb R}^n)$, $g\in L^2({\Bbb R}^n)$. \par 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 $p$. \par 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. \par 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.
[Christian Pötzsche (Minneapolis)]

MSC 2000:: *35B41 Attractors
35K57 Reaction-diffusion equations
35B45 A priori estimates
35K15 Second order parabolic equations, initial value problems

Keywords: measures of noncompactness; polynomial growth

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]