Generalized Strauss conjecture for semilinear wave equations on $\mathbb{R}^3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911883322195968 |
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| author | Wang, Chengbo Zhang, Xiaoran |
| author_facet | Wang, Chengbo Zhang, Xiaoran |
| contents | In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation $$\partial_{t}^2 u-Δ_{\mathbb{R}^3}u=|u|^{p_S}μ(|u|)\ ,u(0,x)=\varepsilon u_0,\ u_t(0,x)=\varepsilon u_1\ ,$$ where $p_S=1+\sqrt{2}$ is the Strauss critical index in $n=3$, and $μ$ is a modulus of continuity. Inspired by Chen, Reissig\cite{Chen_2024} and Ebert, Girardi, Reissig\cite{MR4163528}, we investigate the sharp condition of $μ$ as the threshold between the global existence and blow up with small data. We obtain the almost sharp results in this paper, which in particular disproves the conjecture in \cite{Chen_2024}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Strauss conjecture for semilinear wave equations on $\mathbb{R}^3$ Wang, Chengbo Zhang, Xiaoran Analysis of PDEs 35L71, 35L05, 35B33, 35A0, 35B44 In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation $$\partial_{t}^2 u-Δ_{\mathbb{R}^3}u=|u|^{p_S}μ(|u|)\ ,u(0,x)=\varepsilon u_0,\ u_t(0,x)=\varepsilon u_1\ ,$$ where $p_S=1+\sqrt{2}$ is the Strauss critical index in $n=3$, and $μ$ is a modulus of continuity. Inspired by Chen, Reissig\cite{Chen_2024} and Ebert, Girardi, Reissig\cite{MR4163528}, we investigate the sharp condition of $μ$ as the threshold between the global existence and blow up with small data. We obtain the almost sharp results in this paper, which in particular disproves the conjecture in \cite{Chen_2024}. |
| title | Generalized Strauss conjecture for semilinear wave equations on $\mathbb{R}^3$ |
| topic | Analysis of PDEs 35L71, 35L05, 35B33, 35A0, 35B44 |
| url | https://arxiv.org/abs/2405.12761 |