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Long-time behavior of solutions to nonlinear reaction diffusion equations involving L 1 data. (English) Zbl 1242.35059

The authors investigate the existence of a global attractor for the reaction diffusion problem

u t -Δu+f(u)=ginΩ× + ,u(x,0)=u 0 (x)inΩ,u(x,t)=0onΩ× + ,( RD )

where Ω N , N2, is a smooth bounded domain, u 0 ,gL 1 (Ω) and f is a function of class C 1 satisfying the following conditions: There exist p2 and positive constants l, C 1 and C 2 such that for all s, f ' (s)-l, C 1 |s| p -kf(s)sC 1 |s| p +k and |f ' (s)|C 2 (1+|s| p-2 ).

It is shown that the semigroup {S(t)} t0 generated by this problem possesses a global attractor 𝒜 in L 1 (Ω) which is invariant, compact in L p-1 (Ω)W 0 1,q (Ω) with

q<max{N/(N-1),(2p-2)/p}

and attracting every bounded subset of L 1 (Ω) in the norm of L r (Ω)H 0 1 (Ω) with r[1,+). The proof is done by a decomposition technique combined with a bootstrap argument to establish some regularity results on the solutions.

The decomposition scheme involves the existence and uniqueness of solutions for the original problem (RD) to obtain regularity results for w(x,t)=u(x,t)-v(x) which satisfies

w t -Δw=f(v)-f(v+w)inΩ× + ,w(x,0)=u 0 (x)-v(x)inΩ,w(x,t)=0onΩ× + ,

where v satisfies the elliptic equation -Δv+f(v)=ginΩ with homogeneous Dirichlet boundary condition and vW 0 1,q (Ω) for q<max{(2p-2)/p,N/(N-1)}.

MSC:
35B41Attractors (PDE)
35B65Smoothness and regularity of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
35K57Reaction-diffusion equations
35K58Semilinear parabolic equations