Study of a nonlinear renewal equation with diffusion

Abstract: We consider a nonlinear age structured McKendrick-von Foerster population model with diffusion term (MV-D). We prove the existence and uniqueness of solution (both weak and classical) of the MV-D equation. For the weak solution, we also prove the convergence of the solution to its steady state as time tends to infinity using the generalized relative entropy inequality and Poincare Writinger type inequality. In the case of the classical solution, we also establish that, the solution of MV-D equation converges pointwise to the solution of its steady state equations as time tends to infinity using the method of subsolution and supersolution.

We propose a numerical scheme for the linear MV-D equation. We discretize the time variable to get a system of second order ordinary differential equations. Convergence of the scheme is established using the stability estimates by introducing Rothe’s function.