Existence Results for a Nonlinear Transport Equation with Unbounded Admissible Velocities Space

Abstract

In this paper, we discuss existence results for the nonlinear boundary value problem (1.1)–(1.2) on \({L^{1}}\)-spaces. This problem was already considered in Latrach and Zeghal (J Appl Math Comp 219:1163–1172, 2012) under the hypothesis that the velocity space has finite measure. This condition was used to establish a compactness result necessary to use fixed point theorems. In this work we show that this hypothesis is not necessary and it can be relaxed. Our analysis uses a new measure of weak noncompactness adapted to the problem, the concept of Dunford–Pettis operators and a new version of Darbo’s fixed point theorem.