Details
| Original language | English |
|---|---|
| Pages (from-to) | 1456-1484 |
| Number of pages | 29 |
| Journal | Communications in Partial Differential Equations |
| Volume | 43 |
| Issue number | 10 |
| Publication status | Published - 23 Feb 2019 |
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.
Keywords
- Conical singularities, Long time existence, Maximal regularity, Porous medium equation, Smoothing effect
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Communications in Partial Differential Equations, Vol. 43, No. 10, 23.02.2019, p. 1456-1484.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities
AU - Roidos, Nikolaos
AU - Schrohe, Elmar
N1 - Funding information: We thank Roland Schnaubelt and Christoph Walker for their help and DeutscheForschungsgemeinschaft for support through grant SCHR 319/9-1 within the program“Geometry at Infinity.”
PY - 2019/2/23
Y1 - 2019/2/23
N2 - 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.
AB - 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.
KW - Conical singularities
KW - Long time existence
KW - Maximal regularity
KW - Porous medium equation
KW - Smoothing effect
UR - http://www.scopus.com/inward/record.url?scp=85062108721&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1708.07542
DO - 10.48550/arXiv.1708.07542
M3 - Article
AN - SCOPUS:85062108721
VL - 43
SP - 1456
EP - 1484
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 10
ER -