The

In this talk I will present results on the time asymptotic behavior of the solution of a nonlocal
dispersion problem with absorption in R^N .

There holds that it strongly depends on the bahavior of the initial datum at infinity. To simplify
the presentation I will only discuss the situation in which the initial datum decays as a negative
power of the norm of the variable. This is,

|x|^α.u0(x) → A > 0  as |x| → ∞, α > 0.

The absorption is modeled as a power of the unknow: −u

p with p > 1 and, the time asymptotics

depends on the relation between α, p and the dimension N.
The idea of the talk is to present the results in all cases and, to give a glimps of the ideas behind
the proofs when dispersion is stronger than absorption or they both persist in the time asymptotics.
These results have been obtained in collaboration with Joana Terra, Carmen Cortazar and
Fernando Quiros.

.

 In the present work, we consider the general class of evolution models, named the
generalized Kaup-Kupershmidt-KdV equation, given by
(1) ut + μ∂3xu + α∂5xu + ∂x(γu2x + P(u)) = 0
where P is a polynomial and μ, α and γ are constants. This evolution model includes
many well known models like the Korteweg-de Vries equation, the modified Korteweg-de
Vries equation, and the Kaup-Kupershmit-Korteweg-de Vries equation. The evolution
model considered has no a Hamiltonian structure as happens in many water wave model.
Using the Fourier transform, the existence of solitary wave solutions for this model is
equivalent to find a fixed point, for which we use the standard Picard method choosing
appropriately the initial condition. We also include a brief discussion on the non existence
of solitary wave solutions.