01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. In the qualitative theory of ordinary differential equations, the Liapunov method plays a fundamental role. To use their analogs for the analysis of stability of solutions to parabolic, hyperparabolic, and other nonclassical equations and systems, time-invariant a priori estimates have to be devised for solutions. In this publication only parabolic problems are considered. Here lie, mainly, the problems which have been investigated most thoroughly --- the construction of Liapunov functionals which naturally generalize Liapunov functions for nonlinear parabolic equations of the second order with one spatial variable. The authors establish stabilizing solutions theorems, and the necessary and sufficient conditions of general and asymptotic stability of stationary solutions, including the so-called critical case. Attraction domains for stable solutions of mixed problems for these equations are described. Furthermore, estimates for the number of stationary solutions are obtained.

Table of Contents:
Introduction LOCAL BEHAVIOR OF SOLUTIONS OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS IN THE NEIGHBORHOOD OF A STATIONARY OR PERIODIC SOLUTION The weight Hölder classes and some auxiliary lemmas Bounded solutions of linear parabolic systems Bounded solutions of nonlinear parabolic systems Integral sets of the nonlinear parabolic systems Stability of integral sets Local theorems of existence and continuous dependence on initial data in the Hölder classes of weight functions CONSTRUCTION OF LIAPUNOV'S FUNCTIONALS IN THE CASE OF ONE SPATIAL VARIABLE Liapunov's functionals in the first order The existence condition for Liapunov's functionals A priori estimates of the first derivative Some generalization of the Liapunov functionals concept Liapunov functionals of the second order A priori estimates of the second derivative Liapunov functionals in the neighborhood of a dynamic problem solution THE BEHAVIOR OF SOLUTIONS OF ONE-DIMENSIONAL NONLINEAR PROBLEMS OVER EXTENDED TIME Liapunovs's functionals and asymptotic behavior of solutions for extended time The discrete Liapunov functional Qualitative properties of mixed problem solutions for nonlinear parabolic equations Some examples Some qualitative properties of dissipative boundary-value problems for quasilinear parabolic equations with one spatial variable THE STABILITY CRITERION FOR THE TRIVIAL SOLUTION TO THE MIXED PROBLEM FOR THE SECOND ORDER PARABOLIC EQUATION The stability criterion for the trivial solution to the linear problem The stability criterion of the trivial solution of the linear mixed problem for the second order parabolic equation with time coefficients that are periodic in time Justification of the linearization method for the bounded nonstationary solution of the parabolic equation Stable solution of the Neumann problem THE ATTRACTION DOMAINS OF STABLE STATIONARY OR STABLE PERIODIC SOLUTIONS Some definitions and the preliminary results The greatest and least periodic solutions of the mixed problem The attraction domains of a stable periodic solution The classification of periodic solutions Solutions, periodic in time, of the mixed problems for autonomous parabolic equations ON STABILIZATION OF MIXED PROBLEM SOLUTIONS FOR AUTONOMOUS QUASILINEAR PARABOLIC EQUATIONS Setting of the problem and the preliminary results Stable -limit sets of solutions of the autonomous quasilinear parabolic equation Unstable -limit sets of solutions for the autonomous quasilinear parabolic equation Stabilization of solutions of boundary-value problems and monotone solutions of boundary-value problems APPENDIX Setting of the problem The basic estimates Proof of theorem 2.1. Estimate for the polynomial function a(?) Estimate for the case µ=0 Setting of the model problem Some solution estimates The general theorem on the estimate for solution derivative for the mixed problem The uniform by the regularization parameters derivative estimate for the model problem and its corollaries The existence theorems for the model and basic problems The uniqueness condition Bibliography