Flow invariance for perturbed nonlinear evolution equations

• Main Author: Dieter Bothe
• Format: Article
• Language: English
• Published: Hindawi Limited 1996-01-01
• Series: Abstract and Applied Analysis
• Subjects: Nonlinear evolution equation; time-dependent constraints; viability; reaction-diffusion system; global existence.
• Online Access: http://dx.doi.org/10.1155/S1085337596000231

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.