On a fully nonlinear parabolic equation and the asymptotic behaviour of its solutions

Abstract

The fully nonlinear parabolic problem (P_{\text{}) u t =min{ψ,Δu} for Ω×R + , u=0 for ∂Ω×R + , u(x,0)=u 0 (x) for Ω , occurs in some cases of Bellman's equation of dynamic programming. The author studies questions of asymptotic behavior of strong solutions of (P_{\text{}). He proves that u(⋅,t) converges as t→∞ , to an equilibrium solution, strongly in H 1 0 (Ω). The correct equilibrium solution is individuated when some conditions are met by either u 0 (for example −Δu 0 ≥0 ) or ψ (for example ψ≥0 , Δψ≥0 ). Instrumental to the above treatment is the study of the problem (P_{\text{}) v t −Δβ(x,v)=0 for Ω×R + , β(x,v)=0 for ∂Ω×R + , v(x,0)=v 0 for Ω , where β(x,r)=−min{ψ,−r} (x∈Ω;r∈R). Problem (P_{\text{}) is shown to be well posed in L 1 (Ω). The difficulty here is represented by the fact that β given above does not meet the standard assumptions that insure that −Δβ(⋅) is m -accretive in L 1 (Ω).