A mathematical analysis of a model of structured population. I
type: Item
instance of: scholarly article
title: A mathematical analysis of a model of structured population. I
zbMATH Open document ID: 1261.47062
published in: Differential and Integral Equations
publication date: 2013-04-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.die/1356012370
review text: The author considers a kinetic model of structured populations \[ \frac {\partial f}{\partial t}+\frac {\partial f}{\partial a}+\mu (a,l)f(t,a,l)=P_{\eta }f=\int _{0}^{l}\int _{0}^{a}\eta (a,l,a^{\prime },l^{\prime })f(t,a^{\prime },l^{\prime })\text{d}a^{\prime }\text{d}l^{\prime } \] with initial data in \(L^{1}(\Omega ),\) where \( \Omega =\left \{ (a,l): 0<a<l, \;0<l<l_{2}\right \} \) with ``boundary'' conditions \(f(t,0,.)=K\tilde {f}(t)\), where \(\tilde {f}(t)\:l\in (0,l_2)\rightarrow f(t,l,l)\) and the boundary operator \(K\) is linear and bounded on \(L^1(0,l_{2})\) with norm \( \left \| K\right \| \geqslant 1\). The main concern of the paper is the well-posedness of this equation in \(L^{1}\) spaces in the sense of semigroup theory. Since \(\mu \in L^{\infty }(\Omega )\) and \(P_{\eta }\in \mathcal {L} (L^{1}(\Omega ))\), without loss of generality, the author restricts himself to the case \(\mu =0\) and \(P_{\eta }=0\). (We point out that generation theory is standard when \(\left \| K\right \| <1\) for general divergence free vector fields and the resulting semigroups are contractive; see, e.g., [\textit{W. Greenberg} et al., Boundary value problems in abstract kinetic theory. Basel-Boston-Stuttgart: Birkhäuser Verlag (1987; Zbl 0624.35003)], Chapter 12.) The author shows that, if \(1_{\varepsilon }\) denotes the multiplication operator by the indicator function of the interval \(\left ( 0,\varepsilon \right ) \) and if \(\lim _{\varepsilon \rightarrow 0}\left \| K1_{\varepsilon }\right \| _{\mathcal {L}(L^{1}(\Omega ))}<1\), then the unbounded operator \(- \frac {\partial f}{\partial a}\) on \(L^{1}(\Omega )\) with boundary condition \( f(0,.)=K\tilde {f}\) (where \(\tilde {f}\) is the trace of \(f\) on \( \left \{ (l,l): 0<l<l_{2}\right \}\)) generates a positive semigroup. The proof consists in using a change of unknown which transforms (in an equivalent way) the initial problem into another one with boundary operator with norm \(<1\). This result is …
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 6152275
zbMATH Keywords: transport equation
zbMATH Keywords: structured populations
zbMATH Keywords: non-contractive boundary conditions
author: Mohamed Boulanouar
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6152275