Age-dependent Equations with Non-linear Diffusion

The function u = u(t, a) usually represents the population density of a certain specie at time t > 0 and age a > 0, so that ū(t) in equation (1.4) is the (weighted) total population independent of age. The operator A[ū](t) in equation (1.1) acts for a fixed function ū and time t as a linear (and unbounded) operator on a Banach space E0. In concrete applications, A[ū](t) plays the role of non-linear diffusion. Equation (1.2) reflects the age-boundary conditions depending on the biological context. The main features of equations (1.1)-(1.4) are the non-linear dependence of the operators A and B on the (total) density u. While a great part of the research so far focused on linear diffusion, it is the aim of this paper to present an approach in an abstract setting giving a framework for a larger class of problems of the form (1.1)-(1.4). This will not only provide us with some flexibility in choosing the underlying functional spaces in concrete applications, but also allows us to consider non-linear diffusion and ageboundary conditions that may depend locally or possibly non-locally with respect to time on the density u. The approach applies to general second order time-dependent elliptic operators on a smooth domain Ω ⊂ R, e.g. to operators of the form A[ū](t)w = −∇x · (