Existence of variational solutions to nonlocal evolution equations via convex minimization

2021/12/01 by Prasad, Harsh, Tewary, Vivek · 1 citation
#35A01 #35A15 #35K51 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.00402

Abstract

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation ∂_t u amp;+ P.V.∫_ℝ^N \frac|u(x,t)-u(y,t)|^p-2(u(x,t)-u(y,t))|x-y|^N+ps
amp;+a(x,y)\frac|u(x,t)-u(y,t)|^q-2(u(x,t)-u(y,t))|x-y|^N+qr dy = 0. 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.

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• A variational approach to nonlocal heat equations

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