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arXiv · 2508.07802

Critical exponent for semilinear damped wave equations with weighted nonlinear terms and data from Sobolev spaces of negative order

Dinh Van Duong·Tuan Anh Dao
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Abstract

In this paper, we would like to study the critical exponent for semilinear damped wave equations with the nonlinearity terms of Coulomb-type singularities $|x|^{-\alpha} |u(t,x)|^p$ and the initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\beta}$. Precisely, we obtain a critical exponent $$p_{\rm c}(\alpha,\beta,n): = 1 + \frac{4-2\alpha}{n+2\beta} $$ for $1 \leq n \leq 4$ and $ 0 \leq \alpha, \beta < n/2,$ by proving the global (in time) existence of small data solutions when $p \geq p_{\rm c}(\alpha,\beta,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(\alpha,\beta,n)$. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.

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Dinh Van Duong, Tuan Anh Dao. 2025-08-11. Critical exponent for semilinear damped wave equations with weighted nonlinear terms and data from Sobolev spaces of negative order. https://arxiv.org/abs/2508.07802

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