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On the blow-up for a Kuramoto-Velarde type equation

Oscar Jarrin, Gaston Vergara-Hermosilla

Abstract

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters $γ_1$ and $γ_2$ involved in the non-linear terms verify $ γ_1=\frac{γ_1}{2}$ or $γ_2=0$. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework $γ_2\neq \frac{γ_1}{2}$ and $γ_2\neq 0$, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm.

On the blow-up for a Kuramoto-Velarde type equation

Abstract

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters and involved in the non-linear terms verify or . In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework and , we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm.
Paper Structure (5 sections, 6 theorems, 58 equations)

This paper contains 5 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['thm blow-up']}
  3. Appendix: Proof of technical lemmas
  4. Acknowledgements.
  5. Statements and Declaration

Key Result

Theorem 1.1

Let $u_0$ be the initial datum defined by (Initial-data) and (Condition-eta). Moreover, within the framework of the conditions (Framework), assume that $\gamma_1$ and $\gamma_2$ also verify and with $C_1>0$ big enough. Then, the solution $u(t,x)$ of equation Main-Equation arising from $u_0$ blows-up at the time ${T_*= \frac{2\ln(2)}{3}}$, and for $s>-1$ we have ${\| u(T_*,\cdot)\|_{\dot{H}^s}=+\

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.1
  • Remark 1
  • Lemma 2.2
  • Proposition 2.2
  • Lemma 2.3