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Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

Ahmed Bchatnia, Makram Hamouda, Firas Kaabi, Takiko Sasaki, Hatem Zaag

Abstract

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

Abstract

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable (). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ().

Paper Structure

This paper contains 9 sections, 22 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Preliminary computations
  3. Main results
  4. Proof of Theorem Theorem 1: Construction via Successive Approximations
  5. Blow-up rates of solutions and Lipschitz continuity of T
  6. Linearity of the Blow-up Curve for Limit Solutions
  7. Blow-up Limits and Regularity of the Blow-up Time
  8. Linearity of the Blow-up Curve for Limit Solutions
  9. Proof of the Continuous Differentiability of the Blow-up Curve

Key Result

Lemma 2.1

For initial data satisfying $\gamma_1 + \gamma_2 > A$, where $A = 2\mu^{1/(p-1)}$, the solution of 1.6 exhibits a finite-time blow-up, that is, there exists $T_1 = T_1(\gamma_1,\gamma_2) > 0$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (48)

  • Lemma 2.1: Finite-time blow-up
  • proof
  • Remark 3.1
  • Theorem 3.2: Existence and regularity of blow-up curve
  • Remark 3.3
  • Theorem 3.4: Blow-up rates
  • Theorem 3.5: Lipschitz continuity
  • Theorem 3.6: Linear structure of the limiting blow-up curve
  • Theorem 3.7: $\mathcal{C}^1-$regularity of the blow-up curve
  • Remark 4.1: Preservation of nonnegativity
  • ...and 38 more