Topics in partial differential equations of elliptic and hyperbolic types

This research is focused on partial differential equations and related topics. The results on existence of global solutions of nonlinear damped wave systems and on Sturm type comparison theorems for nonlinear inequalities, with their applications to quasilinear elliptic equations, form the main body of my dissertation.; 1. Quasilinear elliptic equations. The purpose of this part of the work is to extend the classical Sturm theorem for second order linear equations to nonlinear inequalities, and to apply the resulting comparison theorems to study existence and uniqueness problems for positive radially symmetric solutions of quasilinear elliptic equations. Our results extend recent comparison theorems of Gui and Filippucci-Ghiselli Ricci-Pucci.; We focus on qualitative and quantitative relationships between positive solutions of the following inequalities {dollar}{dollar}(psb1(r)phi(u)vert uspprimevertsp{lcub}m-2{rcub}uspprime)spprime + qsb1(r)f(u)ge 0 quad{lcub}rm for{rcub} r > 0eqno(1){dollar}{dollar}and{dollar}{dollar}(psb2(r)phi(v)vert vspprimevertsp{lcub}m-2{rcub}vspprime)spprime + qsb2(r)f(v)le 0 quad{lcub}rm for{rcub} r > 0.eqno(2){dollar}{dollar} Basic models leading to such nonlinear inequalities are the degenerate Laplace equation, the steady state porous medium equation and the conformal Gaussian curvature equation. Our comparison theorems have also been applied in the following contexts: (a) Estimates for the first eigenvalue of the m-Laplace operator on the unit ball in {dollar}IRsp{lcub}n{rcub}{dollar}. (b) Uniqueness of positive solutions for boundary value problems for degenerate Laplace equations. (c) Existence and nonexistence theorems for positive entire solutions of quasilinear elliptic equations. (d) Uniqueness of positive entire radial solutions of quasilinear elliptic equations.; 2. Nonlinear damped wave systems. We study the existence of global solutions for the vector system{dollar}{dollar}usb{lcub}tt{rcub} - Delta u + Q(t,x,u,usb{lcub}t{rcub}) + f(x,u) = 0 rm on lbrack 0, infty) times Omega.eqno(3){dollar}{dollar}Here u is a vector in {dollar}IRsp{lcub}n{rcub}{dollar} with smooth boundary, and Dirichlet boundary conditions are imposed.; The existence of global solutions of the initial-boundary value problem for equations of type (3) has been of considerable interest to mathematicians and physicists. Here we establish an existence theorem for the system (3) which extends the existence theorems of Lions-Strauss and Nakao, even for the case when (3) is an equation ({dollar}N = 1{dollar}).; The method which we develop to study the existence problem for the system (3) can also be applied to degenerate or singular parabolic systems. In particular, an existence theorem for distribution solutions has been obtained for the system {dollar}{dollar}beta(t,u,usb{lcub}t{rcub}) = Delta u + f(x,u) rm on lbrack 0, infty)timesOmega.eqno(4){dollar}{dollar}One of the most important models for (4) is porous medium type equations with nonlinear source terms, that is{dollar}{dollar}usb{lcub}t{rcub} = Delta (vert uvertsp{lcub}m-1{rcub} u) + f(x,u) rm on lbrack 0, infty) timesOmega.{dollar}{dollar}...