Lipschitz regularity for viscosity solutions to parabolic 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-= p(x, t) = p+ < 8 for some constants 1 < p-= p+ < 8. Using Ishii-Lions' method, a Lipschitz estimate of viscosity solutions is established on Riemannian manifolds with sectional curvature bounded from below.