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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 $$\gather {\partial u\over\partial t}= \nabla(G(x, u)\circ\nabla u)- Bu+ F(u_t),\quad x\in\Omega,\quad t\ge t_0,\\ {\partial u\over\partial n}\Biggl|_{\partial\Omega}= 0,\quad t\ge t_0,\\ u(t_0+ s,x)= \phi(s,x)\quad -r\le s\le 0,\quad x\in\Omega,\endgather$$ where $u_t(x)= u(t+ s,x)$, $\phi(s, x)\in C([- r,0]\times \Omega;\bbfR^m)$. By $T(t)\phi= u_t(x)$ the semigroup of operators is defined for $\phi\in C^L= C([- r,0], L^2(\Omega))$. The authors prove that the semigroup $T(t)$ possesses a compact global and connected attractor in $C^L$.

##### MSC:
 35K57 Reaction-diffusion equations 35R10 Partial functional-differential equations 35B40 Asymptotic behavior of solutions of PDE 35B41 Attractors (PDE)
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##### References:
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