Lipschitz regularity for viscosity solutions to parabolic $${\varvec{p(x,t)}}$$-Laplacian equations on Riemannian manifolds

Abstract

We study viscosity solutions to parabolic p(x, t)-Laplacian equations on Riemannian manifolds under the assumption that a continuous exponent function p is Lipschitz continuous with respect to spatial variables, and satisfies $$ 1< p_- \le p(x,t)\le p_+<\infty $$ for some constants $$1<p_-\le p_+ <\infty $$ . Using Ishii–Lions’ method, a Lipschitz estimate of viscosity solutions is established on Riemannian manifolds with sectional curvature bounded from below.