Qualitative Theory of Parabolic Equations Part 1 1st Edition by Zelenyak, Vishnevskii, Lavrentiev 9067642363 978-9067642361

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Product details:

ISBN 10: 9067642363

ISBN 13: 978-9067642361

Author: T. I. Zelenyak; M. P. Vishnevskii; M. M. Lavrentiev

Table of contents:

1 Local behavior of solutions of boundary-value problems for nonlinear parabolic systems in the neighborhood of a stationary or periodic solution
§1.1 The weight Hölder classes and some auxiliary lemmas
§1.2 Bounded solutions of linear parabolic systems
§1.3 Bounded solutions of nonlinear parabolic systems
§1.4 Integral sets of the nonlinear parabolic systems. Stability of integral sets
§1.5 Local theorems of existence and continuous dependence on initial data in the Hölder classes of weight functions
2 Construction of Liapunov’s functionals in the case of one spatial variable
§2.1 Liapunov’s functionals of the first order
§2.2 The existence condition for Liapunov’s functionals
§2.3 A priori estimates of the first derivative
§2.4 Some generalization of the Liapunov functionals concept
§2.5 Liapunov functionals of the second order
§2.6 A priori estimates of the second derivative
§2.7 Liapunov functionals in the neighborhood of a dynamic problem solution
3 The behavior of solutions of one-dimensional nonlinear problems over extended time
§3.1 Liapunov’s functionals and asymptotic behavior of solutions for extended time
§3.2 The discrete Liapunov functional
§3.3 Qualitative properties of mixed problem solutions for nonlinear parabolic equations
§3.4 Some examples
§3.5 Some qualitative properties of dissipative boundary-value problems for quasilinear parabolic equations with one spatial variable
4 The stability criterion for the trivial solution to the mixed problem for the second order parabolic equation
§4.1 The stability criterion for the trivial solution to the linear problem
§4.2 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
§4.3 Justification of the linearization method for the bounded nonstationary solution of the parabolic equation
§4.4 Stable solution of the Neumann problem
5 The attraction domains of stable stationary or stable periodic solutions
§5.1 Some definitions and the preliminary results
§5.2 The greatest and least periodic solutions of the mixed problem
§5.3 The attraction domains of a stable periodic solution
§5.4 The classification of periodic solutions
§5.5 Solutions, periodic in time, of the mixed problems for autonomous parabolic equations
6 On stabilization of mixed problem solutions for autonomous quasilinear parabolic equations
§6.1 Setting of the problem and preliminary results
§6.2 Stable ω-limit sets of solutions of the autonomous quasilinear parabolic equation
§6.3 Unstable ω-limit sets of solutions for the autonomous quasilinear parabolic equation
§6.4 Stabilization of solutions of boundary-value problems and monotone solutions of boundary-value problems
Appendix
§A.1 Setting of the problem
§A.2 The basic estimates
§A.3 Proof of theorem 2.1
§A.4 Estimate for the polynomial function
§A.5 Estimate for the case μ ≠ 0
§A.6 Setting of the model problem. Some solution estimates
§A.7 The general theorem on the estimate for solution derivative for the mixed problem
§A.8 The uniform by the regularization parameters derivative estimate for the model problem and its corollaries
§A.9 The existence theorems for the model and basic problems. The uniqueness condition

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Tags: Zelenyak, Vishnevskii, Lavrentiev, Qualitative