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

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: $$u_{tt}-u_{xx}+\frac{\mu }{t}{u}_t=|u{|}^p,t>t_0,x\in \mathbb{R}.$$ 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 $$u_{tt}-\Delta u+\frac{\mu }{t}{u}_t=|u{|}^p,t>t_0,x\in {\mathbb{R}}^n,$$ with powers $p$ greater than Fujita exponent and sufficiently large $\mu$.

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Cited by in corpus (5)

• Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity
• A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime
• On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity
• Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case
• Global existence of solutions for semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime