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Existence results for a class of nonlinear singular transport equations in bounded spatial domains
Author(s) -
Latrach Khalid,
Oummi Hssaine,
Zeghal Ahmed
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5995
Subject(s) - mathematics , bounded function , operator (biology) , fixed point theorem , mathematical analysis , nonlinear system , bounded operator , measure (data warehouse) , collision , domain (mathematical analysis) , compact space , finite rank operator , boundary (topology) , compact operator , pure mathematics , banach space , computer science , physics , quantum mechanics , extension (predicate logic) , biochemistry , chemistry , computer security , repressor , database , transcription factor , gene , programming language
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.