Asymptotic properties of solutions to the Cauchy problem for degenerate\n parabolic equations with inhomogeneous density on manifolds

2020/08/28 by Daniele Andreucci, Andreucci, Daniele, Anatoli F. Tedeev +1 · 2 citations
Mathematics · #advanced mathematical theories #Geometric Analysis and Curvature Flows #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2008.12666

Abstract

We consider the Cauchy problem for doubly nonlinear degenerate parabolic\nequations with inhomogeneous density on noncompact Riemannian manifolds. We\ngive a qualitative classification of the behavior of the solutions of the\nproblem depending on the behavior of the density function at infinity and the\ngeometry of the manifold, which is described in terms of its isoperimetric\nfunction. We establish for the solutions properties as: stabilization of the\nsolution to zero for large times, finite speed of propagation, universal bounds\nof the solution, blow up of the interface. Each one of these behaviors of\ncourse takes place in a suitable range of parameters, whose definition involves\na universal geometrical characteristic function, depending both on the geometry\nof the manifold and on the asymptotics of the density at infinity.\n

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