On large solutions for fractional Hamilton-Jacobi equations

TL;DR

This paper investigates the existence and blow-up behavior of large solutions for nonlocal fractional Hamilton-Jacobi equations with fully nonlinear elliptic operators, revealing new phenomena due to the nonlocal diffusion.

Contribution

It introduces the analysis of large solutions for fractional Hamilton-Jacobi equations with nonlocal operators, highlighting novel blow-up phenomena absent in local cases.

Findings

Existence of large solutions with boundary blow-up behavior.

Identification of new blow-up phenomena involving solutions tending to -infinity.

Differences in solution behavior between nonlocal and local cases.

Abstract

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical power $0<p<2s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0<p<2s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry-Lions for the case subquadratic case $1<p<2$.