Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation

Firas Kaabi

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arXiv · 2605.01010

Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation

Firas Kaabi

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Abstract

We consider a strongly damped semilinear wave equation with initial data prescribed as $(\varrho\phi,\varrho h)$, where the profiles are fixed and only the amplitude $\varrho>0$ is allowed to vary. The question addressed here is how this rescaling affects a guaranteed lower bound for the maximal existence time. We show that the solution exists at least on a time interval of length comparable to $\varrho^{-(p-2)}$. The proof is based on the growth of a quadratic phase-space norm: after the source term is estimated by the relevant Sobolev embedding, the problem reduces to a scalar differential inequality. The constants produced in the argument are independent of $\varrho$, so the dependence on the initial amplitude remains explicit throughout.

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Firas Kaabi. 2026-05-01. Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation. https://arxiv.org/abs/2605.01010

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