Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

TL;DR

This paper investigates the loss of classical boundary conditions in solutions to fully nonlinear parabolic equations with superquadratic gradient growth, showing finite-time boundary condition failure under certain initial data conditions.

Contribution

It demonstrates the nonexistence of global classical solutions under specific initial data and boundary conditions, highlighting the loss of boundary conditions in finite time.

Findings

Solutions can lose boundary conditions in finite time.

Global solutions exist in viscosity sense but not classically.

Boundary conditions are not preserved for large initial data.

Abstract

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, that a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.