The optimal initial datum for a class of reaction–advection–diffusion equations. (August 2022)

• Record Type: Journal Article
• Authors: Abdul Halim, Omar; El Smaily, Mohammad
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: 35K55 -- 35K57 -- 35K58; Optimal initial data -- Optimization in semilinear parabolic equations -- Maximal mass -- Enhancement by advection; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112877
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a reaction–diffusion model with a drift term in a bounded domain. Given a time T, we prove the existence and uniqueness of an initial datum that maximizes the total mass ∫ Ω u ( T, x ) d x in the presence of an advection term. In a population dynamics context, this optimal initial datum can be understood as the best distribution of the initial population that leads to a maximal the total population at a prefixed time T . We also compare the total masses at a time T in two cases: depending on whether an advection term is present in the medium or not. We prove that the presence of a large enough advection enhances the total mass.