On the semilinear damped wave equation with Riesz potential-type power nonlinearity and initial data in pseudo-measure spaces
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| Format: | Preprint |
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2026
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| author | Loc, Tang Trung Van, Duong Dinh An, Phan Duc |
| author_facet | Loc, Tang Trung Van, Duong Dinh An, Phan Duc |
| contents | In this paper, our main objective is to determine the critical exponent for the semilinear damped wave equation with Riesz potential-type power nonlinearity $\mathcal{I}_γ(|u|^p)$ for $γ\geq 0$, and initial data belonging to the pseudo-measure spaces $\mathcal{Y}^q$. Our main approach is to establish decay estimates for solutions to the corresponding linear problem in the $L^2$-framework with initial data belonging to $\mathcal{Y}^q$, combined with some tools from Harmonic Analysis. Consequently, we derive a new critical exponent $$p_{\mathrm{crit}}(n, q, γ):=1+\frac{2+γ}{n-q},$$ for $1 \leq n \leq 4$ and $0 \leq γ< q < n/2$, by proving the global (in time) existence of small data solutions when $p \geq p_{\mathrm{crit}}(n, q, γ)$, and blow-up of weak solutions in finite time, even for small initial data, whenever $1<p<p_{\mathrm{crit}}(n, q, γ)$. Moreover, we are going to provide sharp estimates for the lifespan of solutions when a blow-up phenomenon occurs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_16293 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the semilinear damped wave equation with Riesz potential-type power nonlinearity and initial data in pseudo-measure spaces Loc, Tang Trung Van, Duong Dinh An, Phan Duc Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 In this paper, our main objective is to determine the critical exponent for the semilinear damped wave equation with Riesz potential-type power nonlinearity $\mathcal{I}_γ(|u|^p)$ for $γ\geq 0$, and initial data belonging to the pseudo-measure spaces $\mathcal{Y}^q$. Our main approach is to establish decay estimates for solutions to the corresponding linear problem in the $L^2$-framework with initial data belonging to $\mathcal{Y}^q$, combined with some tools from Harmonic Analysis. Consequently, we derive a new critical exponent $$p_{\mathrm{crit}}(n, q, γ):=1+\frac{2+γ}{n-q},$$ for $1 \leq n \leq 4$ and $0 \leq γ< q < n/2$, by proving the global (in time) existence of small data solutions when $p \geq p_{\mathrm{crit}}(n, q, γ)$, and blow-up of weak solutions in finite time, even for small initial data, whenever $1<p<p_{\mathrm{crit}}(n, q, γ)$. Moreover, we are going to provide sharp estimates for the lifespan of solutions when a blow-up phenomenon occurs. |
| title | On the semilinear damped wave equation with Riesz potential-type power nonlinearity and initial data in pseudo-measure spaces |
| topic | Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 |
| url | https://arxiv.org/abs/2602.16293 |