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

Abstract

Abstract. We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness and maximal L p -regularity of a short time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin-Sobolev spaces and R-sectoriality perturbation techniques.

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

“…Step 2 : For σ > (n + 1)/p and γ > (n + 1)/2, the space H σ,γ p (B) embeds into the Zygmund space C σ−(n+1)/p * (B); this is shown e.g. in the proof of [20,Corollary 3.3]. In view of Corollary 4.7 and (5.28) we see that the elements of V are uniformly Hölder continuous for some small Hölder exponent.…”

“…It is well known, see e.g. [20], that the domain of the minimal extension of ∆, i.e. the closure of ∆ considered as an operator on C ∞ c (B • ), differs from that of the maximal extension, which consists of all u ∈ H s,γ p (B) such that ∆u ∈ H s,γ p (B), by a finite dimensional space E. In order to understand this space, recall that the conormal symbol σ M (∆) of ∆ is the operator-valued function…”

“…The interval I γ will then include the pole 0 = q − 0 (but none of the q − j for j ≥ 1), and the space E will contain the space E 0 of all smooth functions which are locally constant near ∂B. As in [20] we therefore focus on the closed extension ∆ s with the domain…”

The harmonic map heat flow on conic manifolds

“…The first embeddings follow from [6, Lemma 5.4] and [22, Lemma 3.6]. The second embeddings were proved in [22,Lemma 5.2]. The extra assumption γ + 2θ − ε > n+1 2 is to guarantee the sums of the weighted Mellin Sobolev spaces with C are direct.…”

“…In this subsection, we will quote some well-established results on the closed extensions of the conic Laplace-Beltrami operator. More details of these results can be found in [21,22,23]. We denote by 0 = λ 0 > λ 1 > · · · the distinct eigenvalues of ∆ B and by E 0 , E 1 , · · · the corresponding eigenspaces.…”