Existence, uniqueness and regularity for nonlinear parabolic equations with nonlocal terms

Abstract

We study evolution equations that are fully nonlinear, degenerate parabolic, nonlocal and nonmonotone. The major difficulty lies in nonmonotonicity, i.e. in the fact that no comparison principle can be obtained. This implies that the classical method used to prove existence in the context of fully nonlinear degenerate equations, namely Perron’s one, does not apply. We thus need to use a fixed point argument and to get suitable a priori estimates, we need a refined version of classical continuous dependance estimates. This technical result is of independent interest. We also obtain results such as uniform or Hölder continuity of the solution.