## ABSTRACT

This chapter is devoted to the Monge-Ampère equation which is a celebrated example of an equation which can change character. It introduces functionals which are good candidates for producing Liar punov functions for some hyperbolic P.D.E.s. The chapter is also devoted to an hyperbolic heat conduction equation and to Arnold’s theorem concerning the nonlinear stability of steady plane flows of an ideal incompressible fluid. It performs a complete qualitative analysis of a nonlinear integro-differential equation of the particle transport theory. The chapter considers an I.B.V.P. for an exterior, elastic region where the elastic field is allowed to grow rapidly at infinity and derive therefor some estimates of the continuous dependence type. It discusses other cross-sectional estimates for problems of the initial boundary value type modified in respect of the smoothness required of the solution and in respect of the global characterization of initial data.