Nonlinear Evolution Equations

This book explains several methods for investigation of the existence of solutions of nonlinear evolution equations and asymptotic properties of global solutions. The first chapter has introductory outlines and contains also a list of important evolution equations. Linear contraction semigroups are briefly introduced in chapter 2 and their properties are used in proving the existence results for semilinear equations. The compactness method and the method of monotone operators are described in chapter 3. It is shown in chapter 4 how the comparison principle can be used for the convergence of monotone iterations. Construction of invariant regions is also explained in this section. The methods given in chapters 2 -- 4 generally yield solutions that can blow up in finite time. The problem of the existence of small global solutions is examined in chapter 5. In particular, a priori estimates in Lp-norms are presented here. Chapter 6 is devoted to the convergence of global solutions to stationary ones and the existence of global attractors. Since the book concentrates on the main features of describing methods and leaves out various technical generalizations it is readable and is recommended mainly to graduate students in various fields of nonlinear science with a good background in mathematics.

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