Asynchronous exponential growth for an age dependent population equation with delayed birth process (Q816398)

This work deals with the qualitative and quantitative study of a population equation with age structure and delayed birth process. More precisely, under some conditions on the death rate \(\mu\) and the birth rate \(\beta\), the authors prove first the well-posedness of the following population equation \[ \begin{cases}\partial_t u(t,a)=-\partial_a u(t,a)-\mu(a)u(t,a), &t\geq 0,\,\,a\geq 0,\cr u(t,0)= \int_0^\infty \int_{-\tau}^0 \beta(\sigma ,a)u(t+\sigma ,a)\,d\sigma da,&t\geq 0,\cr u(s,a)=F(s,a),&s\in [-\tau ,0),\,\,a>0,\cr u(0,a)=f(a),&a\geq0. \end{cases} \] Moreover, they show that the solution has asynchronous exponential growth. The techniques used are operator matrices, Hille-Yosida operators and spectral theory of positive semigroups.