Nonlinear Partial Differential Equations, 2010
Asymptotic Behavior of Solutions and Self-Similar Solutions
Progress in Nonlinear Differential Equations and Their Applications Series, Vol. 79

The purpose of this book is to present typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial di?erential equations of di?usion type. For instance, we examine such eq- tions by analyzing special so-called self-similar solutions. We are in particular interested in equations describing various phenomena such as the Navier? Stokesequations.Therescalingmethod describedherecanalsobeinterpreted as a renormalization group method, which represents a strong tool in the asymptotic analysis of solutions of nonlinear partial di?erential equations. Although such asymptotic analysis is used formally in various disciplines, not seldom there is a lack of a rigorous mathematical treatment. The intention of this monograph is to ?ll this gap. We intend to develop a rigorous mat- matical foundation of such a formalasymptotic analysis related to self-similar solutions. A self-similar solution is, roughly speaking, a solution invariant under a scaling transformationthat does not change the equation. For several typical equations we shall give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions. Since nonlinear partial di?erential equations are used not only in mat- matics but also in various ?elds of science and technology, there is a huge variety of approaches. Moreover,even the attempt to cover only a few typical ?elds and methods requires many pages of explanations and collateral tools so that the approaches are self-contained and accessible to a large audience.

Asymptotic Behavior of Solutions of Partial Differential Equations.- Behavior Near Time Infinity of Solutions of the Heat Equation.- Behavior Near Time Infinity of Solutions of the Vorticity Equations.- Self-Similar Solutions for Various Equations.- Useful Analytic Tools.- Various Properties of Solutions of the Heat Equation.- Compactness Theorems.- Calculus Inequalities.- Convergence Theorems in the Theory of Integration.

Challenges the reader with many exercises, examples and illustrations Includes recent developments and several open problems Serves as an excellent textbook for a first course in modern analysis or as an excellent resource for a self-study guide Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented Includes supplementary material: sn.pub/extras