Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

TL;DR

This paper investigates boundary condition attainment for a fractional Laplacian-based parabolic equation, revealing conditions for classical boundary adherence and finite-time boundary loss depending on initial data size.

Contribution

It establishes existence of classical boundary solutions for small times and certain initial data, and demonstrates finite-time boundary condition loss for larger initial data.

Findings

Classical boundary solutions exist for small times with sufficiently regular initial data.

Boundary condition loss occurs in finite time depending on initial data size.

The problem is well-posed globally when boundary data is interpreted in the viscosity sense.

Abstract

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are \emph{a)} the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all H\"older-continuous initial data, with H\"older exponent above a critical a value, and \emph{b)} the nonexistence of solutions satisfying the boundary data in the classical sense for all time. In this case, the phenomenon of loss of boundary conditions occurs in finite time, depending on a largeness condition on the initial data.