A strong maximum principle for nonlinear nonlocal diffusion equations

TL;DR

This paper establishes a strong maximum principle for a class of nonlinear nonlocal diffusion equations, showing that solutions attaining interior extrema are trivial, with implications for the structure of solutions and well-posedness.

Contribution

It introduces a maximum principle for nonlocal nonlinear diffusion equations applicable to bounded functions, and characterizes trivial solutions, including nonconstant discontinuous cases.

Findings

Solutions with interior extrema are trivial.

Trivial solutions may be discontinuous and nonconstant.

Problems are globally well-posed for Lipschitz, nonnegative diffusion coefficients.

Abstract

This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions. Solutions that attain interior global extrema must be identically trivial. However, depending on the nonlinearity, trivial solutions may not be constant in space; they may have an infinite number of discontinuities, for example. We give examples of nonconstant trivial solutions for different nonlinearities. For porous medium-type equations, these functions do not solve the associated classical differential equations, even those in weak form. We also show that these problems are globally wellposed for Lipschitz, nonnegative diffusion coefficients.