On parabolic systems with discontinuous nonlinearities

Abstract

We consider parabolic systems

over (O,T)×Ω with bounded but discontinuous nonline-arities. Here A₁,A₂ are positive elliptic operators of order 2 m with continuous coefficients, f is a bounded function having a jump in u=1, and g₁, g₂ are Lipschitz continuous and bounded. We prescribe Dirichlet boundary conditions and the initial values u(O)=φ¹, v(O)=φ². Essentially under the condition f(u _o) =O we then prove the existence of global solutions which are almost regular everywhere, i.e.: u′ , v′ ϵ L^p(Ω)), u,v ϵ L ^p ((O,T), H^2m,p(Ω)) for some large p and for all T > O and consequently λ^2m−1u, λ^2m−1v are Hölder continuous in t, x over . Our proof is based on the construction of a putative solution (u,v) by approximation and the study of the set u =u_o. Although we have in mind applications to second order equations with A₁ =A₂=−λ we purposely work within a more general framework; thereby we want to show that we need neither the maximum principle nor any monotonicity properties of the nonlinear part.

AMS(MOS):

• 35K55