Flow invariance for perturbed nonlinear evolution equations

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.

Keywords

• Nonlinear evolution equation
• time-dependent constraints
• viability
• reaction-diffusion system
• global existence.