A theoretical analysis of certain types of nonlinear evolution equations with applications

Abstract

The aim of this study was to examine and analyse the existence results of a class of nonlinear evolution equations that describe various phenomena from different areas such as Biology, Physics and Chemistry. First, the global dynamics of a coupled system of partial differential equations with ordinary differential equations modelling an SVEIR epidemic model with age-dependent vaccination was examined by constructing a Lyapunov functionals and application of Lasalle's invariance principle. Next, the solvability of a nonlinear non-autonomous integro-differential equation describing coagulation-fragmentation processes with growth was investigated using a modified monotone method. Existence and uniqueness of results were obtained thanks to Gronwall inequality. In particular, a new concept of upper-lower solution was introduced and a comparison principle established. Finally, the global existence of weak solutions of a nonlinear system, consisting of a differential equation, coupled with a non-autonomous integro-differential equation describing the dynamic of prion proliferation was established by employing a weak compactness method. It is assumed that polymers can split into two or more pieces at a certain rate that not only depends on the sizes of the polymers involved but also on time. The degradation and splitting rates were also considered to be unbounded.