Flow Invariance for Perturbed Nonlinear Evolution Equations

Let X be a real Banach space, J = [0, a] ⊂ IR, A : D(A) ⊂ X → 2 \ ∅ an m-accretive operator and f : J × X → X continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets K ⊂ X for the evolution system u′ +Au f(t, u) on J = [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 = ∆Φ(u) + g(u) in (0,∞)× Ω, Φ(u(t, ·)) ∣∣ ∂Ω = 0, u(0, ·) = u0 under certain assumptions on the set Ω ⊂ IR the function Φ(u1, . . . , um) = (φ1(u1), . . . , φm(um)) and g : IR+ → IR.