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

Marcello D'Abbicco

TL;DR

The study investigates global-in-time energy solutions for the semilinear Euler-Poisson-Darboux equation with a critical time-dependent damping \mu/t and a power nonlinearity. It develops a sharp linear theory via a fundamental solution expressed with Bessel functions and comprehensive frequency-localized L^r-L^q estimates, then leverages a contraction mapping to prove global existence for small data when p > p_c = \max\{p_0(1+\mu),3\}. In one dimension, the results reveal a transition governed by the shifted Strauss exponent and Fujita exponent, while in higher dimensions a diffusion-dominated regime requires larger \mu and p above the Fujita-type threshold; extensions to Tricomi-type generalized equations and L^2-based weak solutions are also provided. The findings clarify how dissipation strength shapes global existence thresholds, contribute explicit decay and energy estimates, and offer a rigorous framework for analyzing nonlinear wave-diffusion transitions in related models.

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: $$u_{tt}-u_{xx} + \fracμ{t}\,u_t = |u|^p \,, \quad t>t_0, \ x\in\mathbb{R}\,.$$ 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 $μ$.

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

TL;DR

The study investigates global-in-time energy solutions for the semilinear Euler-Poisson-Darboux equation with a critical time-dependent damping \mu/t and a power nonlinearity. It develops a sharp linear theory via a fundamental solution expressed with Bessel functions and comprehensive frequency-localized L^r-L^q estimates, then leverages a contraction mapping to prove global existence for small data when p > p_c = \max\{p_0(1+\mu),3\}. In one dimension, the results reveal a transition governed by the shifted Strauss exponent and Fujita exponent, while in higher dimensions a diffusion-dominated regime requires larger \mu and p above the Fujita-type threshold; extensions to Tricomi-type generalized equations and L^2-based weak solutions are also provided. The findings clarify how dissipation strength shapes global existence thresholds, contribute explicit decay and energy estimates, and offer a rigorous framework for analyzing nonlinear wave-diffusion transitions in related models.

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: Here either (singular problem) or (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 , but also on the parameter . We prove that, assuming small initial data in and in the energy space, global-in-time energy solutions exist for , for any , where is the critical exponent for the semilinear wave equation without dissipation in space dimension , conjectured by W.A. Strauss, and 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 with powers greater than Fujita exponent and sufficiently large .

Paper Structure

This paper contains 23 sections, 12 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. The criticality of the dissipative term
  3. The influence of the dissipation on the critical exponent
  4. The transition to a new critical exponent
  5. Result for the one-dimensional case
  6. Result in the multidimensional case
  7. Notation
  8. Estimates for the linear problem
  9. The fundamental solution to \ref{['eq:CPlinear']}
  10. $L^r-L^q$ estimates at low frequencies
  11. $L^{r_1}-L^q$ estimates at intermediate frequencies
  12. $L^{r_2}-L^q$ estimates at high frequencies
  13. The case of nonnegative integer values of $\nu$
  14. Unifying the estimates at different frequencies
  15. Energy estimates
  16. ...and 8 more sections

Key Result

Theorem 1

Let $n=1$, $\mu>0$ and $p>p_{\mathrm{crit}}=\max\{p_{0}(1+\mu),3\}$. Then there exists $\varepsilon>0$ such that for any there exists a unique $u\in\mathcal{C}([0,\infty),H^1)\cap\mathcal{C}^1([0,\infty),L^2)$, global-in-time energy solution to eq:CPsing, and for any there exists a unique $u\in\mathcal{C}([t_0,\infty),H^1)\cap\mathcal{C}^1([t_0,\infty),L^2)$, global-in-time energy solution to eq

Theorems & Definitions (35)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Theorem 3
  • Remark 1.4
  • Remark 1.5
  • Definition 1
  • Theorem 4
  • ...and 25 more