A scalar transport equation

Abstract

Here the variables x, y, t are non-negative, and the functions f(x, 0), 4(x, y) and #1(x, y) are assumed to be known. The main result (Theorem 1) is that under certain hypotheses on f(x, 0), 4(x, y) and yG(x, y) there exists a continuous solution f(x, t), valid for x, t_ 0, which is non-negative, analytic in t for each x, and integrable in x for each f. Another hypothesis guarantees uniqueness. A special form of equation (1), with 1 0, was treated from a practical point of view in [3]. More recently an existence theorem has been proved by Morgenstern [4], which applies to a general class of equations including the case ImO of equation (1). The method of proof used in the present paper applies not only to equation (1) but also to certain other equations of the form