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

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.

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…In this last section, we will study the well-posedness and various properties of (8.1) for m > 0. These results generalize the study of the porous medium equation on manifolds with singularities in [43,44,50].…”

“…We regard ∆ as a cone differential operator or a Fuchs type operator and recall some basic facts and results from the related underlined pseudodifferential theory, which is called cone calculus, towards the direction of the study of nonlinear partial differential equations. For more details we refer to [6], [13], [14], [25], [28], [36], [37], [38], [39], [40], [41], [42], [43], [44] and [45].…”

“…Concerning the situation of the usual porous medium equation, the problem has already been considered on spaces with non-trivial geometry; we briefly mention the following contributions. In [38] it was shown existence, uniqueness and maximal L q -regularity for the short time solutions, where in [39] this result was improved to long time existence and smoothness. Moreover, concerning the case of singular manifolds in the sense of H. Amann [1], in [46] it was shown existence, uniqueness and maximal continuous regularity for the short time solutions and in [47] global existence of L 1 -mild solutions; see also [48] and [49] for similar problems on such spaces.…”

Expanding solutions of quasilinear parabolic equations

“…In addition, in [15,Theorem 4.4] a necessary and sufficient condition was found such that the above solution exists for all times. By combining the results in [15] with [17,Theorem 3.1] we obtain the following smoothness for the solutions of (4.13)- (4.14). A(Λ t1 (π/2); D(∆ k r ))…”

Closedness and invertibility for the sum of two closed operators

“…for showing existence of long time solutions for quasilinear parabolic problems (see e.g. [15,Theorem 4.6]).…”