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

Feng, Tingfu; Li, Ruixi; Yang, Mengrui

doi:10.30511/ttmp.2025.2061423.1054

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