Examples of shocks in population dynamics (Q2815964)

Substantially the paper deals with a problem for a linear hyperbolic partial differential operators of first order in \(\mathbb{R}^{2}\)\ of the following type NEWLINE\[NEWLINE\begin{cases}\frac{\partial u}{\partial t}(t,x)+\frac{\partial u}{\partial x}(t,x)=-\mu (x)u(t,x), \\ u(0,x)=\varphi (x), \\ u(t,0)=\int\limits_{0}^{x_{m}}\beta (y)u(t,y)dy, \end{cases}NEWLINE\]NEWLINE where the functions \(\mu ,\varphi \) and \(\beta \) are given data. The main results of the formulated problem are obtained in the framework of generalized functions of Colombeau type when the data \(\varphi \) and \(\beta \) are Sobolev-Schwartz distributions.NEWLINENEWLINEThe problem studied is illustrated by the mathematical formulation of a model in demography.