Nonlocal Harnack inequalities for nonlocal heat equations

Abstract

By applying the De Giorgi-Nash-Moser theory, we obtain nonlocal Harnack inequalities for locally non-negative weak solutions of nonlocal parabolic equations given by an integro-differential operator L-K as follows: {L(K)u + partial derivative(t)u = 0 in Omega(I) := Omega x (-T, 0] u = g in partial derivative(p)Omega(I) : = ((R-n\Omega) x (-T, 0]) boolean OR(Omega x {t = -T}) for g is an element of C(R-I*(n)) boolean AND L-infinity (R-n x (-T, 0]) boolean AND H-T(s)(R-n) and a bounded domain Omega subset of R-n with Lipschitz boundary. Interestingly, this result implies the classical Harnack inequalities for globally nonnegative weak solutions. (C) 2019 Elsevier Inc. All rights reserved.