Flow invariance for perturbed nonlinear evolution equations. (English) Zbl 0954.34054

The author’s abstract: “Let \(X\) be a real Banach space, \(J=[0,a] \subset \mathbb{R}\), \( A : D(A) \subset X \to 2^X \) an \(m\)-accretive operator and \(f : J\times X \to X\) a continuous function. The author obtains a sufficient condition for weak positive invariance (also called viability) of closed sets \(K \subset X\) for the evolution system \( u' + Au \ni f(t,u)\) on \( J=[0,a]\). More generally, he provides conditions under which this evolution system has a mild solution satisfying time-dependent constraints \(u(t) \in K(t)\) on \(J\). This result is then applied to obtain global solutions to reaction-diffusion systems with nonlinear diffusion, e.g. of type \(u_t = \triangle \Phi (u) + g(u)\) in \((0,\infty) \times \Omega\), \(\Phi(u(t, \cdot))|_{\partial \Omega} =0\), \(u(o, \cdot) + \infty\) under certain assumptions on the set \(\Omega \subset \mathbb{R}^n\), the function \(\Phi(u_1, \dots ,u_m) = (\varphi (u_1), \dots, \varphi (u_m))\) and \(g: \mathbb{R}_+^m \to \mathbb{R}^m\)”.


34G25 Evolution inclusions
35K57 Reaction-diffusion equations
37C27 Periodic orbits of vector fields and flows
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