Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

TL;DR

This paper establishes the existence of weak solutions for a broad class of nonlocal, nonlinear second-order parabolic equations, motivated by front propagation, dislocation theory, and Fitzhugh-Nagumo systems.

Contribution

It provides new existence results for complex nonlocal nonlinear parabolic equations, extending previous theories to include applications in front propagation and biological systems.

Findings

Existence of weak solutions demonstrated for the class of equations.

Applicable to front propagation, dislocation theory, and Fitzhugh-Nagumo models.

Broadens mathematical understanding of nonlocal nonlinear parabolic PDEs.

Abstract

In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.