论文标题
半线性Euler-Poisson-Darboux方程：临界耗散的波案例
The semilinear Euler-Poisson-Darboux equation: a case of wave with critical dissipation
论文作者
论文摘要
在本文中，我们研究了Euler-Poisson-Darboux方程的Cauchy问题的存在，具有功率非线性：$$ u_ {tt} -u_ {xx} + \fracμ{t} + \fracμ{t} x \ in \ mathbb {r} \，。$$这里$ t_0 = 0 $（单数问题）或$ t_0> 0 $（常规问题）。该模型代表具有关键耗散的波方程，从某种意义上说，拥有全局小数据解决方案的可能性不仅取决于功率$ p $，还取决于参数$μ$。我们证明，假设在$ l^1 $中以少量的初始数据，并且在能源空间中，对于$ p> p_c = \ max \ {p_0（1+μ），3 \} $，对于任何$μ> 0 $，$ p_0（k）$是半偏移$ k $ k $ k $ k $ k $ k $ k $ k. 0 $> 0 $，其中$ p_0（k）是至关重要的。 $ 3 $是H. Fujita获得半线性热方程的关键指数。我们还为多维EPD方程$$ U_ {tt}-ΔU + \fracμ{t} \，u_t = | U | U | U |^p \ ,, quad T> t_0，\ x \ in \ mathbb {r}^n \ p powers和powers和powers和powers和powers和powers和powers和powers和powers和足够大的$μ$。
In this paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$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 $μ$.
