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arXiv 2008.08703math.AP
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半线性Euler-Poisson-Darboux方程：具有临界耗散的波动情形

The semilinear Euler-Poisson-Darboux equation: a case of wave with critical dissipation

Marcello D'Abbicco

发表信息：
Journal of Differential Equations 286 (2021) 531--556
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AI总结：

本文研究半线性Euler-Poisson-Darboux方程（具有临界耗散）的Cauchy问题，证明在初始数据小且 $p>\max\{p_0(1+\mu),3\}$ 时存在整体能量解，并给出多维情形在 $\mu$ 足够大时的结果。

摘要：

本文研究了具有幂非线性项的Euler-Poisson-Darboux方程的Cauchy问题整体能量解的存在性：$$u_{tt}-u_{xx} + \frac{\mu}{t}\\,u_t = |u|^p \\,, \quad t>t_0, \\ x\in\mathbb{R}\\,.$$ 这里 $t_0=0$（奇异问题）或 $t_0>0$（正则问题）。该模型代表具有临界耗散的波动方程，因为存在全局小数据解的可能性不仅取决于幂 $p$，还取决于参数 $\mu$。我们证明，假设初始数据在 $L^1$ 和能量空间中是小量，则对于任意 $\mu>0$，当 $p>p_c =\max\{p_0(1+\mu),3\}$ 时，存在整体时间能量解，其中 $p_0(k)$ 是W.A. Strauss猜想的一维无耗散半线性波动方程的临界指数，而 $3$ 是H. Fujita获得的半线性热方程的临界指数。我们还收集了多维EPD方程 $$u_{tt}-\Delta u + \frac{\mu}{t}\\,u_t = |u|^p \\,, \quad t>t_0, \\ x\in\mathbb{R}^n\\,,$$ 在幂 $p$ 大于Fujita指数且 $\mu$ 足够大时的小数据解的整体存在性结果。

英文摘要：

In this paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$u_{tt}-u_{xx} + \fracμ{t}\,u_t = |u|^p \,, \quad t>t_0, \ x\in\mathbb{R}\,.$$ Here either $t_0=0$ (singular problem) or $t_0>0$ (regular problem). This model represents a wave equation with critical dissipation, in the sense that the possibility to have global small data solutions depend not only on the power $p$, but also on the parameter $μ$. We prove that, assuming small initial data in $L^1$ and in the energy space, global-in-time energy solutions exist for $p>p_c =\max\{p_0(1+μ),3\}$, for any $μ>0$, where $p_0(k)$ is the critical exponent for the semilinear wave equation without dissipation in space dimension $k$, conjectured by W.A. Strauss, and $3$ is the critical exponent obtained by H. Fujita for semilinear heat equations. We also collect some global-in-time existence result of small data solutions for the multidimensional EPD equation $$u_{tt}-Δu + \fracμ{t}\,u_t = |u|^p \,, \quad t>t_0, \ x\in\mathbb{R}^n\,,$$ with powers $p$ greater than Fujita exponent and sufficiently large $μ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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