Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914525969645568 |
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| author | Kaabi, Firas |
| author_facet | Kaabi, Firas |
| contents | We consider a strongly damped semilinear wave equation with initial data prescribed as $(\varrhoϕ,\varrho h)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01010 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation Kaabi, Firas Analysis of PDEs 35L05, 35B44, 35B40, 35L71 We consider a strongly damped semilinear wave equation with initial data prescribed as $(\varrhoϕ,\varrho h)$, 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. |
| title | Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation |
| topic | Analysis of PDEs 35L05, 35B44, 35B40, 35L71 |
| url | https://arxiv.org/abs/2605.01010 |