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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.18677 |
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| _version_ | 1866913938687393792 |
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| author | Qianqian, Li Dinghuai, Wang Huicheng, Yin |
| author_facet | Qianqian, Li Dinghuai, Wang Huicheng, Yin |
| contents | There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and $p_f(n)=1+\frac{2}{n}$. It is noticed that the weak solution $u$ can blow up in finite time when $1<p\le p_{crit}(n,μ)$. In addition, for $n=1$, this open question has been solved recently. We now systematically solve this open problem for $n=2$. As the first part, in the present paper, the global small solution $u$ is established for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. Our main ingredients are to find the suitable conformal power $p_{conf}(2,μ)$ and derive some new kinds of spacetime-weighted $L^{q}_tL^{q}_x([1, \infty)\times \mathbb{R}^2)$ or $L^q_tL^ν_rL^2_θ([1, \infty)\times [0, \infty)\times [0, 2π])$ Strichartz estimates for the solutions of linear generalized Tricomi equation $\partial_t^2v-t^mΔv=F(t,x)$ ($m>0$). In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $p>1$ and $μ>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I Qianqian, Li Dinghuai, Wang Huicheng, Yin Analysis of PDEs There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and $p_f(n)=1+\frac{2}{n}$. It is noticed that the weak solution $u$ can blow up in finite time when $1<p\le p_{crit}(n,μ)$. In addition, for $n=1$, this open question has been solved recently. We now systematically solve this open problem for $n=2$. As the first part, in the present paper, the global small solution $u$ is established for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. Our main ingredients are to find the suitable conformal power $p_{conf}(2,μ)$ and derive some new kinds of spacetime-weighted $L^{q}_tL^{q}_x([1, \infty)\times \mathbb{R}^2)$ or $L^q_tL^ν_rL^2_θ([1, \infty)\times [0, \infty)\times [0, 2π])$ Strichartz estimates for the solutions of linear generalized Tricomi equation $\partial_t^2v-t^mΔv=F(t,x)$ ($m>0$). In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $p>1$ and $μ>0$. |
| title | Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.18677 |