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

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 L p ‐spaces with 1 ≤  p <+ ∞ . This problem was already considered in 6,8,9 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 betweenX p σ(a weighted space) andX p (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.