Flow invariance for perturbed nonlinear evolution equations

Dieter Bothe

Abstract and Applied Analysis, 1996, vol. 1, 1-17

Abstract:

Let X be a real Banach space, J = [ 0 , a ] ⊂ R , A : D ( A ) ⊂ X → 2 X \ 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 ′ + A u ∍ 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 u t = Δ Φ ( u ) + g ( u ) in ( 0 , ∞ ) × Ω , Φ ( u ( t , ⋅ ) ) | ∂ Ω = 0 , u ( 0 , ⋅ ) = u 0 under certain assumptions on the set Ω ⊂ R n the function Φ ( u 1 , … , u m ) = ( φ 1 ( u 1 ) , … , φ m ( u m ) ) and g : R + m → R m .

Date: 1996
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:950983

DOI: 10.1155/S1085337596000231

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