Blow-up results for a logarithmic pseudo-parabolic $p(.)$-Laplacian type equation

TL;DR

This paper investigates blow-up phenomena and decay behavior for a variable exponent pseudo-parabolic equation with logarithmic nonlinearity, providing criteria for blow-up and conditions for global existence.

Contribution

It introduces new blow-up criteria and decay estimates for solutions of a complex variable exponent pseudo-parabolic PDE with logarithmic nonlinearity.

Findings

Established a blow-up criterion using a differential inequality method.

Derived an upper bound for the blow-up time.

Showed decay estimates under small initial data conditions.

Abstract

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla u\right\vert^{p(.)-2}\nabla u\right) =|u|^{q(.)-2}u\ln(|u|), \quad (x,t)\in\Omega\times(0,+\infty),$$ where $\Omega\subset\mathbb{R}^n$ is a bounded and regular domain, and the variable exponents $p(.)$ and $q(.)$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(\Omega)$-norm as $t\to+\infty$. This decay result is illustrated by a two-dimensional numerical example.