Communications in Mathematical Sciences
Volume 21 (2023)
High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities
Pages: 25 – 63
In this paper, we consider a class of semilinear parabolic equations with singular potential on manifolds with conical singularities. At high initial energy level $J(u_0) \gt d$, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1.1)-(1.3) respectively. Moreover, by applying the Levine’s concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy $J(u_0) \gt d$, including the upper bound of blowup time. Finally, we show a lower bound of the blowup time and blowup rate for problem (1.1)-(1.3) under arbitrary initial energy level.
finite time blow up, blowup time, parabolic equation, conical singularities, singular potential
2010 Mathematics Subject Classification
35A01, 35D30, 35K20, 35K55
This work was supported by the National Natural Science Foundation of China (12271122), the China Postdoctoral Science Foundation (2013 M 540270), the Fundamental Research Funds for the Central Universities.
The research of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PCE 137/2021, within PNCDI III.
Received 7 June 2021
Received revised 22 March 2022
Accepted 28 March 2022
Published 27 December 2022