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Trajectory attractors for reaction-diffusion systems. (English) Zbl 0894.35010
The authors study the notion of attractor for a general class of reaction diffusion systems, i.e.: $\partial_tu= \alpha\Delta u+ f_0(u, t)+g(x, t)\tag{1}$ with Neumann or Dirichlet boundary conditions. Here $$\alpha$$ is a constant matrix, $$\alpha= (\alpha_{jk})$$, $$j,k\leq n$$, subject to conditions which guarantee ellipticity; $$f$$, $$g$$ are continuous in their arguments, but $$f$$ need not satisfy a Lipschitz condition; thus solutions to (1) must not be unique. In Section 1, the authors describe a general frame adapted to the above situation, Sections 2, 3 are devoted to applications. We content us with a glance at Section 1.
Starting point is the evolution equation $\partial_tu= A_{\sigma(t)}u,\quad t\geq 0,\tag{2}$ where $$A_{\sigma(t)}$$, $$t\geq 0$$ is an operator family from $$E$$ to $$E_0$$, with $$E$$, $$E_0$$ Banach spaces. Here $$\sigma$$ is a continuous mapping from $$R_+$$ to some topological space $$\psi$$. The set $$X_+$$ of $$\sigma\in C(R_+,\psi)$$ is turned into a topological space via a family of local topologies in a standard way. A translation semigroup $$T_h$$, $$h\geq 0$$ acts on $$X_+$$ according to $$(T_h\zeta)(t)= \zeta(t+ h)$$, $$\zeta\in X_+$$. One assumes that $$T_hX_+\subseteq X_+$$, $$h\geq 0$$. The elements $$\sigma\in X_+$$ are called “symbols” of (2). A subset $$\Sigma\subseteq X_+$$ is then selected (the symbol space) such that $$T_h\Sigma\subseteq\Sigma$$, $$h\geq 0$$, $$\Sigma$$ is metrizable under the relative topology of $$X_+$$. In order to choose $$\Sigma$$ suitably, the authors fix $$\sigma_0\in X_+$$ and denote by $$H_+(\sigma_0)$$ the closure of the linear hull of all $$T_h\sigma_0$$, $$h\geq 0$$ with respect to the topology of $$X_+$$. One has $$T_hH_+(\sigma_0)\subseteq H_+(\sigma_0)$$, $$h\geq 0$$. One calls $$\sigma_0$$ translation compact (t-c) if $$H_+(\sigma_0)$$ is compact in $$X_+$$; the intention is to set $$\Sigma= H_+(\sigma_0)$$ for a $$\sigma_0$$ which is t-c. Finally, $$K^+_\sigma$$ is the set of all solutions of (2) for the symbol $$\sigma$$; one sets $$K_+= \bigcup K^+_\sigma$$, $$\sigma\in \Sigma$$. Again one has $$T_hK_+\subseteq K_+$$, $$h\geq 0$$. In a last step, the authors define the notion of trajectory attractor $$A\subseteq K_+$$ with respect to the translation semigroup $$T_h$$, $$h\geq 0$$. This definition is to long to be reproduced here. Theorem 1.1 then asserts under some technical conditions on the $$K^+_\sigma$$ the existence of a trajectory attractor endowed with various properties. As indicated, these notions and results are then applied to concrete systems of type (1).

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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