Chapter 6 Application of Monotone Type Operators to Parabolic and Functional Parabolic PDE's

Abstract

Publisher Summary This chapter discusses the formulation of the main notions and results in the theory of monotone type operators and their application to (possibly nonlinear) parabolic differential equations and parabolic functional differential equations. The abstract Cauchy problem is considered in the chapter for first order evolution equations in a finite interval. The main results on the existence, uniqueness, and continuous dependence of the weak solutions of higher order nonlinear parabolic differential equations are also provided in the chapter. The chapter includes higher order nonlinear functional parabolic equations, where only the lower order terms contain functional dependence. It also details second order nonlinear parabolic functional differential equations, where also the main part contains functional dependence. It discusses the existence and qualitative properties of the solutions of parabolic functional differential equations in (0, ∞) and elaborates further applications of monotone type operators, for example, to the systems of functional parabolic equations. Several examples are also given in the chapter for the “general” results.