In this paper, we prove the existence of a countable family of regular spherically symmetric self-similar solutions to focusing energy-supercritical semi-linear wave equations

$$\begin{aligned} \partial _{tt}u-\Delta u=|u|^{p-1}u \qquad \text {in} \,\, \mathbb {R}^{N}, \end{aligned}$$

where \(N\ge 3\), \(1+\frac{4}{N-2}<p\) and, if \(N\ge 4\), \(p \le 1+\frac{4}{N-3}\). This was previously known only in the case \(N=3\), for integer p (see Bizoń et al. in Nonlinearity 20(9):2061–2074, 2007). We also study the asymptotics of these solutions.

中文翻译：

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$${\mathbb {R}}^{N}$$ RN 中聚焦半线性波动方程的自相似解

在本文中，我们证明了聚焦能量-超临界半线性波动方程的可数族正则球对称自相似解的存在

$$\begin{aligned} \partial _{tt}u-\Delta u=|u|^{p-1}u \qquad \text {in} \,\, \mathbb {R}^{N}, \end{对齐}$$

其中\(N\ge 3\) , \(1+\frac{4}{N-2}<p\)并且，如果\(N\ge 4\) , \(p \le 1+\frac{ 4}{N-3}\)。这之前仅在\(N=3\)的情况下才知道，对于整数p（参见 Bizoń 等人在 Nonlinearity 20(9):2061–2074, 2007）。我们还研究了这些解的渐近性。