Classical and Mild Solutions of the Cauchy Problem for a Mildly Quasilinear Wave Equation with Discontinuous and Distributional Initial Conditions

We consider the Cauchy problem for a one-dimensional mildly quasilinear wave equation in the upper half-plane with initial data having a discontinuity of the first kind at one point. We construct a solution of the Cauchy problem in an implicit analytical form as a solution of integral equations. We study the solvability of the integral equations and the smoothness of their solutions. We prove the uniqueness of the solution and find conditions guaranteeing the existence of classical and mild solutions. We construct the solution of the Cauchy problem with the initial data containing the Dirac delta distribution.