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Asymptotic behavior of a class of reaction–diffusion equations with delays. (English) Zbl 1031.35065

The authors study systems of equations of the form

u t=(G(x,u)u)-Bu+F(u t ),xΩ,tt 0 ,u n| Ω =0,tt 0 ,u(t 0 +s,x)=φ(s,x)-rs0,xΩ,

where u t (x)=u(t+s,x), φ(s,x)C([-r,0]×Ω; m ). By T(t)φ=u t (x) the semigroup of operators is defined for φC L =C([-r,0],L 2 (Ω)). The authors prove that the semigroup T(t) possesses a compact global and connected attractor in C L .

35K57Reaction-diffusion equations
35R10Partial functional-differential equations
35B40Asymptotic behavior of solutions of PDE
35B41Attractors (PDE)