A blow-up criterion for a $ p $-Laplacian-type pseudo-parabolic equation involving singular potential and logarithmic source

Hui Yang; Hui Yang

doi:10.3934/era.2026130

Electronic Research Archive

2026, Volume 34, Issue 5: 2868-2882. doi: 10.3934/era.2026130

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Research article

A blow-up criterion for a $ p $-Laplacian-type pseudo-parabolic equation involving singular potential and logarithmic source

Hui Yang ^,

School of Mathematics, Liaoning Normal University, Dalian 116029, China

Received: 08 January 2026 Revised: 25 February 2026 Accepted: 19 March 2026 Published: 31 March 2026

The main goal of this paper was to develop new skills to investigate the blow-up properties of solutions to an initial-boundary value problem for a $ p $-Laplacian-type pseudo-parabolic equation with singular potential and logarithmic source. By establishing a crucial reverse Sobolev inequality, we derived that the solutions blow up in finite time under a new blow-up criterion and we estimated the lifespan of the weak solutions from both above and below. It is worthy to point out that our blow-up criterion implies that this problem admits finite time blow-up solutions at an arbitrarily high initial energy level. From methods to results, we partially extended some results obtained in recent literatures.
- pseudo-parabolic equation,
- $ p $-Laplacian,
- singular potential,
- logarithmic nonlinearity,
- arbitrarily high initial energy,
- blow-up,
- lifespan
Citation: Hui Yang. A blow-up criterion for a $ p $-Laplacian-type pseudo-parabolic equation involving singular potential and logarithmic source[J]. Electronic Research Archive, 2026, 34(5): 2868-2882. doi: 10.3934/era.2026130

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Abstract

The main goal of this paper was to develop new skills to investigate the blow-up properties of solutions to an initial-boundary value problem for a $ p $-Laplacian-type pseudo-parabolic equation with singular potential and logarithmic source. By establishing a crucial reverse Sobolev inequality, we derived that the solutions blow up in finite time under a new blow-up criterion and we estimated the lifespan of the weak solutions from both above and below. It is worthy to point out that our blow-up criterion implies that this problem admits finite time blow-up solutions at an arbitrarily high initial energy level. From methods to results, we partially extended some results obtained in recent literatures.

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