Qualitative Theory of Parabolic Equations, Part 1

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