Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

TL;DR

This paper establishes the existence of variational solutions for a broad class of doubly nonlinear nonlocal evolution equations, including the double phase equation, using a minimizing movements approach for nonlinear parabolic equations with non-standard growth.

Contribution

It introduces a novel application of the minimizing movements method to doubly nonlinear nonlocal equations with complex growth conditions.

Findings

Proves existence of variational solutions for the class of equations.

Extends the minimizing movements approach to nonlocal, doubly nonlinear PDEs.

Addresses equations with non-standard growth conditions.

Abstract

We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{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+qr}} \,dy = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{align*} We make use of the approach of minimizing movements pioneered by DeGiorgi and Ambrosio and refined by B\"ogelein, Duzaar, Marcellini, and co-authors to study nonlinear parabolic equations with non-standard growth.