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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arXiv · 2609.21256

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 $2<p_-\le p(x) \le p_+<2^*$. 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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BibTeX RIS

Menglan Liao, Mingxue Zhang. 2026-09-18. Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source. https://arxiv.org/abs/2609.21256

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