Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

Nikolaos Roidos; Elmar Schrohe

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arXiv · 1708.07542

Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

Nikolaos Roidos·Elmar Schrohe

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Abstract

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in space and time, where the maximal $L^{q}$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

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Nikolaos Roidos, Elmar Schrohe. 2017-08-24. Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities. https://arxiv.org/abs/1708.07542

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