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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 $$g: \mathbb{R}_+^m \to \mathbb{R}^m$$”.

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