Existence of variational solutions to nonlocal evolution equations via convex minimization

Abstract

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

Local boundedness of variational solutions to nonlocal double phase parabolic equations

“…This notion was extended to quasilinear parabolic equations with non-standard growth in [1]. In the companion paper [27], the present authors have extended the notion to nonlocal parabolic equations satisfying only a growth condition from below through convex minimization of approximating functionals. In particular, it covers the double phase equation that we study in this paper.…”

“…Since the variational solution u may not belong to L q (0, T ; W s ′ ,q (R N )), it is not clear whether Steklov averages can be used to obtain the Caccioppoli inequality. For this reason, we assume that ∂ t u ∈ L 2 (Ω T ), which is guaranteed for timeindependent initial-boundary data, as proved in [27].…”

“…Moreover, in [27], we have shown that variational solutions are parabolic minimizers, which we define below. We shall use the more convenient definition of parabolic minimizers in order to prove local boundedness.…”

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

“…In a recent preprint [56], the latter two authors extended the framework of variational solutions to parabolic fractional equations with time independent initial and boundary data. As an application, the latter two authors also studied the local boundedness of variational solutions to double phase nonlocal parabolic equations in [57].…”

“…The proofs of statements (i), (iv), (v) and (vi) are the same as in [6,Lemma B.2] and [9, Lemma 6.2]. The proofs for (ii) and (iii) are given in the appendix of [56].…”