LOCALIZATION OF AGE-DEPENDENT ANTI-CROWDING POPULATIONS

In this work we prove the existence of solutions and study the localization and nonlocalization of the population in the Gurtin-MacCamy model with age-dependence and diffusion partial derivative rho/partial derivative t + partial derivative rho/partial derivative a = rho u(x))(x) = mu(a, u)rho, rho(x, t, 0) = integral 0 infinity beta(a, u)rho(x, t, a) da, rho(x, 0, a) = rho(0)(x, a) greater than or equal to 0, where u(x, t) = integral 0 infinity rho(x, t, a) da and [GRAPHICS] ho>(x, t, a) da and [GRAPHICS] pling it with a set of overposed initial and boundary conditions. The problem obtained is then reduced to a nonlinear Volterra integral equation of second kind for the unknown memory kernel. Then, via the Contraction Principle, we prove local (in time) existence and uniqueness results. In addition, we show the Lipschitz continuous dependence upon the data. These results also apply to a viscoelastic beam model.