A Viscoelastic Wave Equation with Nonlinear Damping and Supercritical Variable-Exponent Logarithmic Sources
Abstract
We investigate a viscoelastic wave equation with nonlinear variable-exponent damping and a variable-exponent logarithmic source, allowing supercritical growth. The main difficulties arise from the failure of the source to map the energy space into L2(Ω), the sign-changing logarithmic potential, and the lack of Sobolev control of the displacement for supercritical damping. Under an exponent condition linking the source and damping exponents, we prove local existence, an energy identity, and a continuation criterion. We prove finite-time blow-up for nontrivial solutions with nonpositive shifted initial energy. We also establish a blow-up criterion based on a sufficiently large initial displacement--velocity pairing, with a threshold depending on the positive part of the shifted initial energy, and derive an explicit upper bound for the lifespan. This criterion allows us to construct blow-up data at every prescribed level of the original energy. For initial data in a suitable stable region, solutions exist globally. For damping within the Sobolev range, a general relaxation condition yields energy decay estimates that include exponential and polynomial rates. Beyond this range, a time-weighted displacement estimate and a splitting of the memory integral give logarithmic decay.
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BibTeX
@misc{liao2026viscoelastic,
title = {{A Viscoelastic Wave Equation with Nonlinear Damping and Supercritical Variable-Exponent Logarithmic Sources}},
author = {Menglan Liao and Mingxue Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15528},
eprint = {2610.15528},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15528},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Viscoelastic Wave Equation with Nonlinear Damping and Supercritical Variable-Exponent Logarithmic Sources
AU - Menglan Liao
AU - Mingxue Zhang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15528
UR - https://arxiv.org/abs/2610.15528
AB - We investigate a viscoelastic wave equation with nonlinear variable-exponent damping and a variable-exponent logarithmic source, allowing supercritical growth. The main difficulties arise from the failure of the source to map the energy space into L^2(Ω), the sign-changing logarithmic potential, and the lack of Sobolev control of the displacement for supercritical damping. Under an exponent condition linking the source and damping exponents, we prove local existence, an energy identity, and a continuation criterion. We prove finite-time blow-up for nontrivial solutions with nonpositive shifted initial energy. We also establish a blow-up criterion based on a sufficiently large initial displacement--velocity pairing, with a threshold depending on the positive part of the shifted initial energy, and derive an explicit upper bound for the lifespan. This criterion allows us to construct blow-up data at every prescribed level of the original energy. For initial data in a suitable stable region, solutions exist globally. For damping within the Sobolev range, a general relaxation condition yields energy decay estimates that include exponential and polynomial rates. Beyond this range, a time-weighted displacement estimate and a splitting of the memory integral give logarithmic decay.
ER -