RT Journal Article
T1 On a fully nonlinear parabolic equation and the asymptotic behaviour of its solutions
A1 Díaz Díaz, Jesús Ildefonso
AB 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).
PB Elsevier
SN 0022-247X
YR 1983
FD 1983-08
LK https://hdl.handle.net/20.500.14352/64669
UL https://hdl.handle.net/20.500.14352/64669
NO United States Army
DS Docta Complutense
RD 1 may 2024