Initial boundary value problem of a class of pseudo-parabolic Kirchhoff equations with logarithmic nonlinearity

TL;DR

This paper investigates the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity, establishing criteria for global existence, blow-up, decay rates, and connections to stationary solutions.

Contribution

It introduces a threshold for global existence and blow-up using the potential well method, and explores the relationship between stationary and global solutions.

Findings

Threshold for global existence and blow-up established.

Exponential decay rate of global solutions determined.

Convergence between ground state and global solutions shown.

Abstract

In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. We use the potential well method to give a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy $J(u_0)\leq d$. When the initial energy $J(u_0)>d$, we find another criterion for the vanishing solution and blow-up solution. We also get the exponential decay rate of the global solution and life span of the blow-up solution. Meanwhile, we study the corresponding stationary problem and establish a convergence relationship between its ground state solution and the global solution.