Local boundedness of variational solutions to nonlocal double phase parabolic equations

TL;DR

This paper establishes local boundedness of solutions to a class of nonlocal double phase parabolic equations, extending regularity results to equations with variable nonlocal exponents and measurable coefficients.

Contribution

It proves local boundedness for variational solutions of nonlocal double phase equations with variable exponents and measurable coefficients, a novel extension in nonlocal PDE regularity theory.

Findings

Solutions are locally bounded under specified conditions.

The results apply to equations with variable nonlocal exponents.

The work extends known regularity results to more general nonlocal operators.

Abstract

We prove local boundedness of variational solutions to the double phase equation \begin{align*} \partial_t u +& P.V.\int_{\mathbb{R}^N}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\ &+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qs'}} \,dy = 0, \end{align*} under the restrictions $s,s'\in (0,1),\, 1 < p \leq q \leq p\,\frac{2s+N}{N}$ and the non-negative function $(x,y)\mapsto a(x,y)$ is assumed to be measurable and bounded.