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PreprintCC BY 4.0arXiv

Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source

  • Menglan Liao
  • Mingxue Zhang
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Abstract

We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies 2<p-≤ p(x) ≤ p+<2*. An optimal pointwise correction to the source--potential inequality defines a \emph{shifted energy}. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.

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Cite

BibTeXRIS
BibTeX
@misc{liao2026blowup,
  title = {{Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source}},
  author = {Menglan Liao and Mingxue Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21256},
  eprint = {2609.21256},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21256},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source
AU  - Menglan Liao
AU  - Mingxue Zhang
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21256
UR  - https://arxiv.org/abs/2609.21256
AB  - We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies 2<p_-≤ p(x) ≤ p_+<2^*. An optimal pointwise correction to the source--potential inequality defines a \emph{shifted energy}. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.
ER  -