Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Asymptotic behaviors of solutions to a singular non-local fourth-order parabolic equation with gradient-type logarithmic nonlinearity

Document Type : Original Article

Authors
Tingfu Feng 1
Ruixi Li 2
Mengrui Yang 1
1 Department of Mathematics, Kunming University, Kunming, 650214, China.
2 Department of Mathematics, Kunming University, Kunming, 650214,China.
10.30511/ttmp.2025.2061423.1054
Abstract
In this paper, we consider the initial-boundary value problem to a singular non-local fourth-order parabolic equation with gradient-type logarithmic nonlinearity. We prove the global existence of weak solutions by utilizing the cut-off function technique and Galerkin's approximation method. We establish novel threshold criteria for the finite-time blow-up of solutions with initial value at arbitrary energy levels, and derive explicit upper bounds for the blow-up time under appropriate conditions. Furthermore, by employing energy estimates and specific ordinary differential inequalities, we characterize both non-extinction and extinction phenomena of solutions in finite time, and rigorously quantify their corresponding extinction rates.
Keywords
Singular non-local fourth-order parabolic equation
Global existence
Blow-up
Non-extinction
Extinction
Subjects
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Transactions in Theoretical and Mathematical Physics
Volume 2, Issue 2
Spring 2025
Pages 91-109

  • Receive Date 22 May 2025
  • Accept Date 30 May 2025
  • First Publish Date 30 May 2025
  • Publish Date 01 May 2025