A mathematical study for a Rotenberg model (Q5957736)

Assume that \(f=f(\mu,\nu,t)\) is the density of a cell population, where \(\mu\) is the degree of maturity, and \(\nu\) is the velocity of maturation. \textit{M. Rotenberg} [J. Theor. Biol. 103, No. 2, 181-199 (1983)] presented a model for \(f\): NEWLINE\[NEWLINE(1)\qquad \partial f/\partial t=-\nu\partial f/\partial \muNEWLINE\]NEWLINE with boundary condition NEWLINE\[NEWLINE(2)\qquad f(t,0,\nu)=\alpha f(t,1,\nu)+p\nu^{-1}\int_a^bk(\nu,\nu')\nu' f(t,1,\nu')d\nu',NEWLINE\]NEWLINE where \(k=k(\mu,\nu')\geq 0\) and NEWLINE\[NEWLINE(3)\qquad \int_a^bk(\nu,\nu')d\nu=1.NEWLINE\]NEWLINE Here, \(\nu\) and \(\nu'\) express daughter cell and mother cell, respectively, and \(0\leq p\) is the average number of viable daughters per mitotic in the degenerate case.NEWLINENEWLINENEWLINEThe author is concerned with the existence of a strongly continuous semigroup of the model (1)--(3) for all \(\alpha\geq 0\) and all \(p\geq 0\). Some properties, such as positivity, irreducibility, and asymptotic behavior in the uniform topology are also investigated. New techniques are introduced into the discussion.