Existence of variational solutions to nonlocal evolution equations via convex minimization

¹ Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore 560065, Karnataka, India
² School of Interwoven Arts and Sciences, Krea University, Sri City 517646, India

^* Corresponding author: vivek.tewary@krea.edu.in

Received: 6 September 2022
Accepted: 13 December 2022

Abstract

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation

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.