Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities

Pages: 25 – 63

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a2

Authors

Yuxuan Chen (School of Mathematical Science, Heilongjiang University, Harbin, China; College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, China)

Vicenţiu D. Rădulescu (Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland; and Department of Mathematics, University of Craiova, Romania)

Runzhang Xu (College of Mathematical Sciences, Harbin Engineering University, Harbin, China)

Abstract

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.

Keywords

finite time blow up, blowup time, parabolic equation, conical singularities, singular potential

2010 Mathematics Subject Classification

35A01, 35D30, 35K20, 35K55

Full Text (PDF format)

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