Existence of variational solutions to nonlocal evolution equations via convex minimization

TL;DR

This paper establishes the existence of variational solutions for a class of nonlocal evolution equations, including the double phase equation, using convex minimization methods that require minimal assumptions.

Contribution

It introduces a novel minimization approach for nonlocal evolution equations, broadening the scope of solvable problems with minimal convexity and coercivity conditions.

Findings

Existence of variational solutions for nonlocal evolution equations proven.

Method applies to double phase equations with minimal assumptions.

Provides a new approach for approximating nonlocal evolution equations.

Abstract

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \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. \end{align*} The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations.