Delta Shocks as Solutions of Conservation Laws with Discontinuous Moving Source

A Riemann problem for the conservation law \(u_{t}+[\phi (u)]_{x}=kH(x-vt)\), where x, t, k, v and \(u=u(x,t)\) are real numbers, is studied with the goal of getting singular solutions in a convenient space of distributions that contains delta shock waves. Here \(\phi \) stands for an entire function taking real values on the real axis and H represents the Heaviside function. When u is seen as a density of matter some surprises may appear such as the creation of matter from a vacuum state. In a particular case, as the time goes on, such a matter grows continuously, running away from any spatial bounded region, what can be viewed as a unidimensional model of universe.