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 $\gamma_1$ and $\gamma_2$ involved in the non-linear terms verify $ \gamma_1=\frac{\gamma_1}{2}$ or $\gamma_2=0$. 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 $\gamma_2\neq \frac{\gamma_1}{2}$ and $\gamma_2\neq 0$, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm.