Nonlinear Partial Differential Equations

In this chapter we turn our attention to the study of the dynamical properties of solutions of nonlinear partial differential equations. We are especially interested here in those nonlinear evolutionary equations which arise in the analysis of two broad classes of partial differential equations: parabolic evolutionary equations and hyperbolic evolutionary equations. While our usage of the terms “parabolic” and “hyperbolic” in this context is motivated by related concepts arising in the basic classification of partial differential equations, we will attribute these terms, instead, to certain dynamical features of the linear ancestry of the underlying nonlinear problems. More precisely, the linear prototypes of the partial differential equations of interest here include the heat equation and the wave equation: $${\partial _t}u - \nu \Delta u = 0\;and\;\partial _t^2u - \nu \Delta u = 0,$$ on a suitable domain Ω in ℝd, with various boundary conditions and initial conditions.