Energy decay for classes of nonlocal dispersive equations.

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: {∂t u + Lαu + u∂x u = 0, x, t ∈ R,u(x, 0) = u0(x), where α > 0, and the operator Lα is the Fourier multiplier operator by a real-valued odd function belonging to (C 1(R) ∩ C ∞(R∗)).

These classes contain, in particular, the following equations: the fractional KdV, Benjamin-Ono and the Intermediate Long Wave, for example.

Posted on Aug 4, 2023 in Differential Equations, Seminars