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

TL;DR

This paper investigates the existence of global solutions for a wave equation with critical dissipation, identifying conditions on the nonlinearity power and dissipation parameter that ensure solutions persist over time.

Contribution

It establishes new existence results for global solutions of the Euler-Poisson-Darboux equation with critical dissipation, extending previous work to include the influence of the dissipation parameter.

Findings

Global solutions exist for nonlinear power p above a critical threshold depending on dissipation and dimension.

The critical exponent depends on the maximum of a wave-related and heat-related critical value.

Small initial data in L^1 and energy space lead to global solutions under specified conditions.

Abstract

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 $\mu$.