Titulo:  Asymptotic description for the localized solution of the Cauchy problem for the  wave equation with fast-oscillating coefficient.
Palestrante: Sergey Sergeev (PUC-Rio)
Data: 23/08/2023
Horário: 12h
Sala: C-119
 
Resumo: We consider the Cauchy problem with localized initial conditions for the multidimensional wave equation. The coefficient of this wave equation is assumed to be fast-oscillating. We are interested in the asymptotic (while localization parameter of initial condition is small) description of the given Cauchy problem. Such formulation leads to the appearance of two small parameters: the localization parameter and the parameter of oscillating in the wave equation coefficient. The ratio between the given parameters is crucial and affects the form of the main part of the asymptotic solution. We use  the homogenization procedure which takes into account this ratio and as result we obtain the equation with smooth coefficients. This equation is of the form of the wave equation with dispersion correction, which appears due to the homogenization procedure. The main part of the asymptotic solution for the initial Cauchy problem thus can be described with the help of the asymptotic solution of the homogenized equation with the smooth coefficients  and can be presented in the analytical form with the help of the Airy functions and related to them.