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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 n (n3) of the form

u t =Δu-λu-f(u)+f x i i +g(x)in + × n (*)

with initial data

u(0,x)=u 0 (x)in n (**)

is shown. Here, the nonlinearity f is allowed to have polynomial growth of arbitrary order p-1 (p2) and in the inhomogeneous term, f x i i , i=1,...,n, are distributional derivatives of fL 2 ( n ), gL 2 ( n ).

In order to obtain this for the problem (*), (**) 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 ( n )-topology without restriction on p.

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

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

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