Existence and uniqueness of solutions for a diffusion model of host--parasite dynamics. (Q1874608)

\textit{F. A. Milner} and \textit{C. A. Patton} [J. Comput. Appl. Math. 154, 273--302 (2003; Zbl 1014.92043)] introduced a new approach to modeling host-parasite dynamics through a convection diffusion partial differential equation, which uses the parasite density as a continuous structure variable. In the present work for this model which is described by a diffusion equation with convective transfer the solvability of the initial boundary value problem is studied. For the class of solutions introduced by the authors under some restrictions on the smoothness properties of the coefficients of the equation based on the estimates of solutions, the solvability of the considered problem is proved. The proof of existence and uniqueness of the solution is based on the technique of comparison of solutions. The large-time behavior of the solution is established.