On the existence of maximal and minimal solutions for parabolic partial differential equations

Abstract

The existence of maximal and minimal solutions for initial-boundary value problems and the Cauchy initial value problem associated with L u = f ( x , t , u , ∇ u ) Lu = f(x,t,u,\nabla u) where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows f to be highly nonlinear, i.e., f locally Hölder continuous with almost quadratic growth in | ∇ u | |\nabla u| .