Flow invariance for perturbed nonlinear evolution equations

Dieter Bothe

Abstract and Applied Analysis, 1996, vol. 1, issue 4, 417-433

Abstract: Let X be a real Banach space, J = [0, a] ⊂ R, A : D(A) ⊂ X → 2X\ϕ 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 = ΔΦ(u) + g(u) in (0, ∞) × Ω, Φ(u(t, ⋅))|∂Ω = 0, u(0, ⋅) = u0 under certain assumptions on the setΩ ⊂ Rn the function Φ(u1, …, um) = (φ1(u1), …, φm(um)) and g:R+m→Rm.

Date: 1996
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:1:y:1996:i:4:p:417-433

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