Energy decay for classes of nonlocal dispersive equations

Speaker: Ricardo Freire

DIM CMM – Universidad de Chile

Date: Monday, August 7th, 2023 at 12 Santiago time

Abstract: We consider the long-time dynamics of large solutions to a special class of evolution equations. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of interior regions, we prove decay for a sequence of times. The classes of nonlocal dispersive equations which we will treat are as follows:

$$\begin{cases} \partial_t u + L_\alpha u + u\partial_x u=0, \quad x,t\in \mathbb{R}, \\u(x,0)=u_0(x)\end{cases}$$

where \(\alpha>0\),and the operator \(L_\alpha\) is the Fourier multiplier operator by a real-valued odd function belonging to \((C^1(\mathbb{R})\cap C^\infty(\mathbb{R}^∗))\). These classes contain, in particular, the following equations: the fractional KdV, Benjamin-Ono and the Intermediate Long Wave, for example.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju