Y. Giga, S. Goto, H. Ishii, M.-H. Sato, Comparison Principle and Convexity Preserving Properties for Singular Degenerate Parabolic Equations on Unbounded Domains, Indiana Univ. Math. J. 40 (1991), 443-470
Abstract
We prove comparison theorems for viscosity solutions of singular degenerate parabolic equations of general form in a domain not necessarily bounded. We also prove that the concavity of solutions is preserved as time develops under additional assumptions on the equations. Both results apply to various equations including the mean curvature flow equation where every level set of solutions moves by its mean curvature.
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