On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ?u/?t + u ?u/?x + u/x = 0 for x > 0, t ? 0, with u(x, 0) = u⁰_?(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰_?(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰_? and u⁰₊, a global shock front weak solution u(x, t) = u_?(x, t) for x < ?(t), u(x, t) = u₊(x, t) for x > ?(t), where u_? and u₊ are the strong solutions corresponding (respectively) to u⁰_? and u⁰₊ and the curve t → ?(t) is defined by d?/dt (t) = 1/2u_?(?(t), t) + u₊(?(t), t)], t ? 0 and ?(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.