Hyperbolic Cauchy problems with singular coefficients

This thesis is devoted to the study of second order weakly hyperbolic equations with singular (less than continuous) coefficients. We first consider equations with time-dependent coefficients and then equations with space-dependent coefficients. We pay particular attention to the role of the lower order terms and we formulate suitable Levi conditions to guarantee the existence of a very weak solution, as introduced in [GR15]. Very weak solutions for these families of equations are also investigated from a numerical point of view in different toy models: the wave equation with a Heaviside function, a delta distribution and homogeneous distributions, respectively, as coefficient in its principal part. The results of this thesis are found in [DGL22a] and [DGL22b].