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Blow-up of solutions to the Euler-Poisson-Darbox equation with critical power nonlinearity

Mengting Fan, Ning-An Lai, Hiroyuki Takamura

Abstract

In our recent precious work, we established the finite time blow up result and upper bound of lifespan estimate to the singular Cauchy problem of semilinear Euler-Poisson-Darboux equation in R^n with subcritical power type nonlinearity. By introducing an improved test function, we obtain an enhanced lower bound for the functional including the spacetime integral of the nonlinear term with an additional logarithmic growth, which finally yields the blow up result and upper bound of lifespan estimate for the corresponding Cauchy problem with "critical" nonlinear power. And this gives some partial answer to the open problem 1 posed by D'Abbicco (J. Differential Equations 286 (2021), 531-556).

Blow-up of solutions to the Euler-Poisson-Darbox equation with critical power nonlinearity

Abstract

In our recent precious work, we established the finite time blow up result and upper bound of lifespan estimate to the singular Cauchy problem of semilinear Euler-Poisson-Darboux equation in R^n with subcritical power type nonlinearity. By introducing an improved test function, we obtain an enhanced lower bound for the functional including the spacetime integral of the nonlinear term with an additional logarithmic growth, which finally yields the blow up result and upper bound of lifespan estimate for the corresponding Cauchy problem with "critical" nonlinear power. And this gives some partial answer to the open problem 1 posed by D'Abbicco (J. Differential Equations 286 (2021), 531-556).
Paper Structure (4 sections, 3 theorems, 100 equations)

This paper contains 4 sections, 3 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Local existence and finite speed propagation
  3. Proof of Theorem \ref{['hddamped']}
  4. Preliminary

Key Result

Theorem 1.2

Let and where Assume the initial data $u_0(x)\in H^1(\mathbf{R}^n)$ is positive and has compact support then there exists a constant $\varepsilon_0=\varepsilon_0(u_0,n,p,\mu,\alpha)>0$ such that the lifespan of the energy solution to the Cauchy problem SHEPD satisfies where $C$ denotes a generic positive constant independent of $\varepsilon$ which may have different values from line to line.

Theorems & Definitions (7)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 3.1
  • Lemma 3.2
  • Proof 3.1