Association Journal of CSIAM
Supervised by Ministry of Education of PRC
Sponsored by Xi'an Jiaotong University
ISSN 1005-3085  CN 61-1269/O1

Chinese Journal of Engineering Mathematics

Previous Articles     Next Articles

Global Existence and Blowup of Solutions for a Class of $p$-Laplace Parabolic Equation

ZHU Liping,  WANG Yan,  YUE Hongyun   

  1. College of Science, Xi'an University of Architecture and Technology, Xi'an 710055
  • Received:2022-05-30 Accepted:2022-09-29 Online:2025-06-15 Published:2025-06-15
  • Supported by:
    The Natural Science Foundation of Shaanxi Province (2020JM-409).

Abstract:

In this paper, by using the potential well method, finite time blowup and global existence of solutions are studied for a class of $p$-Laplace parabolic equations with nonlinear logarithmic term and Neumann boundary conditions. Firstly, the existence of global weak solutions is proved by the Galerkin method and the compactness principle. Then, the decay estimation of the global weak solution is derived by an energy integration and ODE inequality technique. Secondly, through constructing an auxiliary function and combining the concave method, the finite time blowup of the solution under the condition of positive initial energy is obtained,  and the accurate upper bound estimation of blowup time is given for the first time. Furthermore, the results are extended to the case of non-positive initial energy. In order to more intuitively explain the theoretical results, some numerical examples are given to show the long-time decay behavior and the blowup properties of solutions under different initial energy levels, the effects of parameter $ p$ on the evolution of solutions and the correctness of the theoretical results.

Key words: $p$-Laplace, logarithmic nonlinearity, potential well method, global existence, blow up

CLC Number: