Nonlinear Partial Differential Equations

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. 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. This work will serve as an excellent first course in modern analysis. Key topics in nonlinear pde's as well as several fundamental tools and methods are presented; few prerequisites are required of the reader. Challenging exercises, examples, and illustrations help explain the rigorous analytic basis for the Navier-Stokes equations, mean curvature flow equations, and other important equations describing real phenomena. The main focus of the text 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, and in the final chapters recent developments and several open problems are presented. An extensive index is provided.Written for graduate students and researachers by one of Japan's leading analysts, this will be an excellent resource for self-study or classroom use.