Finite-Time Blow-Up for Wave Equations with Nonlinear p-Laplacian Damping
Abstract
We consider a wave equation with linear elasticity, nonlinear p-Laplacian damping acting on the velocity, and a focusing power source in a bounded three-dimensional domain. For 2<p<r+1≤6, we prove finite-time blow-up in the natural energy norm for initial data in H01(Ω)× L2(Ω) lying in the unstable potential well below its depth. This includes every nonzero initial state with nonpositive total energy and requires no sign condition on the initial displacement--velocity pairing. For positive initial energy, we establish a sufficient condition for finite-time blow-up involving the L2 pairing of the initial velocity with a fixed approximation of the initial displacement. The proof combines a shifted displacement multiplier, an auxiliary quantity built from the time integral of the spatial Lp-norm of the velocity gradient, and a weighted auxiliary functional. Linking the weight parameter to the initial value of the cumulative quantity allows us to absorb the mixed terms in the auxiliary functional estimate. Together with the shifted displacement multiplier, this yields upper bounds for the maximal existence time in both ranges 2<p≤3 and p>3, without assuming W01,p(Ω) regularity of the initial displacement.
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BibTeX
@misc{liao2026finitetime,
title = {{Finite-Time Blow-Up for Wave Equations with Nonlinear \(p\)-Laplacian Damping}},
author = {Menglan Liao and Mingxue Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33700},
eprint = {2610.33700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-Time Blow-Up for Wave Equations with Nonlinear p-Laplacian Damping
AU - Menglan Liao
AU - Mingxue Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33700
UR - https://arxiv.org/abs/2610.33700
AB - We consider a wave equation with linear elasticity, nonlinear p-Laplacian damping acting on the velocity, and a focusing power source in a bounded three-dimensional domain. For 2<p<r+1≤6, we prove finite-time blow-up in the natural energy norm for initial data in H_0^1(Ω)× L^2(Ω) lying in the unstable potential well below its depth. This includes every nonzero initial state with nonpositive total energy and requires no sign condition on the initial displacement--velocity pairing. For positive initial energy, we establish a sufficient condition for finite-time blow-up involving the L^2 pairing of the initial velocity with a fixed approximation of the initial displacement. The proof combines a shifted displacement multiplier, an auxiliary quantity built from the time integral of the spatial L^p-norm of the velocity gradient, and a weighted auxiliary functional. Linking the weight parameter to the initial value of the cumulative quantity allows us to absorb the mixed terms in the auxiliary functional estimate. Together with the shifted displacement multiplier, this yields upper bounds for the maximal existence time in both ranges 2<p≤3 and p>3, without assuming W_0^1,p(Ω) regularity of the initial displacement.
ER -