Trajectory attractors for reaction-diffusion systems.

*(English)*Zbl 0894.35010The 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).

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).

Reviewer: B.Scarpellini (Basel)

##### 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 |