Evolution equations with singular coefficients

We study the Cauchy problem for certain classes of evolution equations with singular coefficients and/or data in the framework of the concept of very weak solutions. This concept allows to consider equations with highly singular coefficients and/or data for which the classical theory fail. In particular, it is possible to deal with equations involving the Dirac delta function and its powers as coefficients and/or data. Our study deals with three important questions: The well-posedness of the considered Cauchy problems and the study, either analytically or numerically, of the phenomenon of propagation of coefficients/data singularities. The essential methods for our existence and uniqueness results are based on energy estimates and techniques from the classical analysis of differential equations. In order to describe the behaviour of the very weak solutions near the singularities of the coefficients/data, a detailed phase space analysis is carried out. The approach is based on a decomposition into different zones where different techniques of asymptotic analysis are used.