Lectures on nonlinear evolution equations. Initial value problems. (English) Zbl 0811.35002

Aspects of Mathematics. 19. Braunschweig etc.: Vieweg. viii, 259 p. (1992).
The author investigates the global existence and uniqueness of small smooth solutions for nonlinear evolution equations, including nonlinear wave equation, nonlinear heat equation, nonlinear thermoelastic system etc. The whole book consists of twelve chapters (or sections). In the first ten chapters, the author gives a detailed description about the global existence for the initial value problem for nonlinear wave equation: \(L^ p - L^ q\) decay estimates of solution to linear wave equation; local existence and uniqueness for nonlinear wave equation; a priori estimates of weighted norm of solution; continuation argument.
In Chapter 11, the author briefly discusses global existence and uniqueness of small smooth solution to initial value problem for other nonlinear evolution equations: equations of nonlinear elasticity; nonlinear heat equation; equations of nonlinear thermoelasticity; nonlinear Schrödinger equations; nonlinear Klein-Gordon equations; Maxwell equations and nonlinear plate equations. The basic strategy is still the same: continuation argument by combining local existence with uniform a priori estimates of solutions. In the final chapter, the author shortly discusses further aspects, including the initial boundary value problems and some open problems. This book also describes the contribution of the author to this area, especially to the equations of nonlinear thermoelasticity.
This book systematically describes an important topic in the theory of nonlinear partial differential equations: global existence and uniqueness of small smooth solution. The book is self-contained and well-written. It is worth reading for the readers interested in this topic and the updated developments.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)