We consider the Cauchy problem for the wave  equation with fast-oscillating coefficient and localized initial perturbation in the whole space. This problem contains two small parameters. The first parameter is the localization parameter and it describes the wavelength of the propagating wave. The second parameter 

is the parameter of the fast oscillations of the coefficient.

The global task is to describe the asymptotic solution of the given problem. To do so, we reduce the initial problem to the homogenized one with the equation with variable and smooth coefficients. The localization of the initial conditions allows us to use the semiclassical approximation for the asymptotic description of the solution 

of the homogenized equation.

It turns out that depending on the ratio between two given small parameters the dispersion effects may appear during the asymptotic description of the solution of the homogenized problem.

Our aim in the present talk is to discuss the mechanism of appearance of these dispersion effects and to derive the effective homogenized equations which describe the asymptotic solution with and without dispersion effects.

In this talk we present some recent result about the long time behavior
of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. We prove that when time is large the solution behaves like a gaussian profile on the infinite edges. When the nonlinear convective part is present we obtain similar results but only on a star shaped tree.
Acknowledgment: this is a joint work with Cristian Cazacu (University
of Bucharest), Ademir Pazoto (Federal University of Rio de Janeiro), Julio D. Rossi (University of Buenos Aires) and Angel San Antolin (University of Alicante).