The paper proves short-time existence, uniqueness and maximal L^q-regularity for the fractional porous medium equation on manifolds with conical singularities, and gives the decay rate of solutions near the conical tips.
Smoothness and long time existence for solutions of the Cahn-Hilliard equation on manifolds with conical singularities
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abstract
We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data, we show that the solution exists in the maximal $L^q$-regularity space for all times and becomes instantaneously smooth in space and time, where the maximal $L^q$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.
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The fractional porous medium equation on manifolds with conical singularities I
The paper proves short-time existence, uniqueness and maximal L^q-regularity for the fractional porous medium equation on manifolds with conical singularities, and gives the decay rate of solutions near the conical tips.