Existence results for a class of nonlinear singular transport equations in bounded spatial domains

Abstract

In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on Lp‐spaces with 1 ≤ p<+∞. This problem was already considered in6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between (a weighted space) and (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper.