Lipschitz regularity for viscosity solutions to parabolic p(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{p(x,t)}}$$\end{document}-Laplacian equations on Riemannian manifolds

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+<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1< p_- \le p(x,t)\le p_+<\infty $$\end{document} for some constants 1<p-≤p+<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p_-\le p_+ <\infty $$\end{document}. Using Ishii–Lions’ method, a Lipschitz estimate of viscosity solutions is established on Riemannian manifolds with sectional curvature bounded from below.