Abstract and Applied Analysis
Volume 1 (1996), Issue 4, Pages 417-433

Flow invariance for perturbed nonlinear evolution equations

Dieter Bothe

Fachbereich 17, Universität Paderborn, Paderborn D-33095, Germany

Received 22 November 1996

Copyright © 1996 Dieter Bothe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let X be a real Banach space, J=[0,a]R, A:D(A)X2X\ϕ an m-accretive operator and f:J×XX continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets KX for the evolution system u+Auf(t,u)onJ=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraints u(t)K(t) on J. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type ut=ΔΦ(u)+g(u)in(0,)×Ω,Φ(u(t,))|Ω=0,u(0,)=u0 under certain assumptions on the setΩRn the function Φ(u1,,um)=(φ1(u1),,φm(um)) and g:R+mRm.