Existence and Qualitative Properties of Solutions for Various Quasilinear Systems

Abstract

In this thesis we study a variety of nonlinear partial differential equations (PDE), and systems of PDE. In Chapter 2 we study a general class of coercive, quasilinear elliptic systems which have their origin in the study of viscous, heat-conducting fluids. We pose the problem in an open ball centered at the origin, or all of Euclidean space. In the bounded domain case, we derive sharp conditions for the existence of positive radial solutions with prescribed boundary behavior. In particular, solutions which either blow up at a finite point or remain bounded are considered. We also obtain sharp conditions for the existence of solutions in the whole space, and the exact growth at infinity of such solutions is then determined.

In Chapter 3, we study a prototype reaction-diffusion system. In particular, we focus on the associated steady state problem. Assuming we lie on or above a critical curve, we derive an exact asymptotic expansion of the positive radial solutions. It was previously known that this curve is the threshold for linearly stable radial solutions to exist. We use the asymptotic expansion obtained to prove the nonlinear stability of these solutions, directly extending classical scalar results to the case of systems.

Finally, in Chapter 4 we study a nonlocal quasilinear partial differential inequality which has its origins in non-relativistic mechanics. The nonlocal interactions are modelled using the convolution operation, and a slow decay potential term is considered. Posing the problem in an exterior domain, we derive sharp conditions for the existence and nonexistence of positive solutions. In the case of existence, the optimal decay rate of solutions is also determined. Unlike the cases of fast or critical decay potential terms, exponentially decaying solutions are also shown to exist.