Global existence and blow-up for a fourth-order $ p $-Laplacian dispersive wave equation with logarithmic nonlinearity and strong damping

Guangyang Fu; Jiang Zhou; Guangyang Fu; Jiang Zhou

doi:10.3934/cam.2026036

Communications in Analysis and Mechanics

2026, Volume 18, Issue 4: 931-959. doi: 10.3934/cam.2026036

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Research article

Global existence and blow-up for a fourth-order $ p $-Laplacian dispersive wave equation with logarithmic nonlinearity and strong damping

Guangyang Fu ,
Jiang Zhou ^,

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China

Received: 01 April 2026 Revised: 15 July 2026 Accepted: 16 September 2026 Published: 26 October 2026
35L35, 35D30, 35B44, 35L77

We study the initial boundary value problem for fourth-order strongly damped $ p $-Laplacian dispersive wave equations with logarithmic nonlinearity ($ 1 < p\leq2 $, $ 2\leq r $). By the potential well theory, we prove global solution existence at a subcritical and critical initial energy, and characterize blow-up behaviors: blow-up at infinite time occurs at a subcritical or critical initial energy for $ 1 < p < 2, r = 2 $, and at a supercritical initial energy for $ p = r = 2 $; and for $ 1 < p\leq2, r > 2 $, we establish blow-up at finite time for a subcritical, critical, or supercritical initial energy, and explicit bounds for the blow-up time.
- wave equation,
- blow-up,
- strong damping,
- dispersive,
- logarithmic nonlinearity
Citation: Guangyang Fu, Jiang Zhou. Global existence and blow-up for a fourth-order $ p $-Laplacian dispersive wave equation with logarithmic nonlinearity and strong damping[J]. Communications in Analysis and Mechanics, 2026, 18(4): 931-959. doi: 10.3934/cam.2026036

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Abstract

We study the initial boundary value problem for fourth-order strongly damped $ p $-Laplacian dispersive wave equations with logarithmic nonlinearity ($ 1 < p\leq2 $, $ 2\leq r $). By the potential well theory, we prove global solution existence at a subcritical and critical initial energy, and characterize blow-up behaviors: blow-up at infinite time occurs at a subcritical or critical initial energy for $ 1 < p < 2, r = 2 $, and at a supercritical initial energy for $ p = r = 2 $; and for $ 1 < p\leq2, r > 2 $, we establish blow-up at finite time for a subcritical, critical, or supercritical initial energy, and explicit bounds for the blow-up time.

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