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

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: $$u_{tt}-u_{xx} + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \ x\in\mathbb{R}\,.$$ 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 $\mu$. We prove that, assuming small initial data in $L^1$ and in the energy space, global-in-time energy solutions exist for $p>p_c =\max\{p_0(1+\mu),3\}$, for any $\mu>0$, 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 $$u_{tt}-\Delta u + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \ x\in\mathbb{R}^n\,,$$ with powers $p$ greater than Fujita exponent and sufficiently large $\mu$.