The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts with recent developments and several open problems. With challenging exercises, examples, and illustrations to help explain the rigorous analytic basis for the Navier--Stokes equations, mean curvature flow equations, and other important equations describing real phenomena, this book is written for graduate students and researchers, not only in mathematics but also in other disciplines.

Nonlinear Partial Differential Equations will serve as an excellent textbook for a first course in modern analysis or as a useful self-study guide. Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented. The only prerequisite required is a basic course in calculus.

Klappentext

This work will serve as an excellent first course in modern analysis. The main focus is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. This textbook will be an excellent resource for self-study or classroom use.

Inhalt

Preface.- Part I Asymptotic Behavior of Solutions of Partial Differential Equations.- 1 Behavior Near Time Infinity of Solutions of the Heat Equation.- 2 Behavior Near Time Infinity of Solutions of the Vorticity Equations.- 3 Self-Similar Solutions for Various Equations.- Part II Useful Analytic Tools.- 4 Various Properties of Solutions of the Heat Equation.- 5 Compactness Theorems.- 6 Calculus Inequalities.- 7 Convergence Theorems in the Theory of Integration.- Answers to Exercises.- Comments on Further References.- References.- Glossary.- Index.