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 ${\gamma }_2=\frac{{\gamma }_1}{2}$ 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 ${\gamma }_2\ne \frac{{\gamma }_1}{2}$ and ${\gamma }_2\ne 0$, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm. As a complement to it, we address an alternative result on the finite-time blow-up of smooth solutions by considering a virial-type estimate.