Convergence of Bounded Solutions for Nonlinear Parabolic Equations
Convergence of Bounded Solutions for Nonlinear Parabolic Equations

Author: Hon To Hardy Chan

Publisher:

Published: 2013

Total Pages: 142

ISBN-13:

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Zelenyak proved in 1968 that every bounded global solution of a second order quasilinear parabolic equation converges to a stationary solution. An important part in the proof is that every such equation has a variational structure. For higher order parabolic equations, this is not the case. In this thesis, we prove Zelenyak's theorem and find a necessary and sufficient condition for a fourth or sixth order equation to be variational.

Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations
Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations

Author: V. Lakshmikantham

Publisher: CRC Press

Published: 2003-02-27

Total Pages: 336

ISBN-13: 9780415305280

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A monotone iterative technique is used to obtain monotone approximate solutions that converge to the solution of nonlinear problems of partial differential equations of elliptic, parabolic and hyperbolic type. This volume describes that technique, which has played a valuable role in unifying a variety of nonlinear problems, particularly when combined with the quasilinearization method. The first part of this monograph describes the general methodology using the classic approach, while the second part develops the same basic ideas via the variational technique. The text provides a useful and timely reference for applied scientists, engineers and numerical analysts.

Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems
Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems

Author: Jens Lang

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 161

ISBN-13: 3662044846

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Nowadays there is an increasing emphasis on all aspects of adaptively gener ating a grid that evolves with the solution of a PDE. Another challenge is to develop efficient higher-order one-step integration methods which can handle very stiff equations and which allow us to accommodate a spatial grid in each time step without any specific difficulties. In this monograph a combination of both error-controlled grid refinement and one-step methods of Rosenbrock-type is presented. It is my intention to impart the beauty and complexity found in the theoretical investigation of the adaptive algorithm proposed here, in its realization and in solving non-trivial complex problems. I hope that this method will find many more interesting applications. Berlin-Dahlem, May 2000 Jens Lang Acknowledgements I have looked forward to writing this section since it is a pleasure for me to thank all friends who made this work possible and provided valuable input. I would like to express my gratitude to Peter Deuflhard for giving me the oppor tunity to work in the field of Scientific Computing. I have benefited immensly from his help to get the right perspectives, and from his continuous encourage ment and support over several years. He certainly will forgive me the use of Rosenbrock methods rather than extrapolation methods to integrate in time.

Linear and Nonlinear Parabolic Complex Equations
Linear and Nonlinear Parabolic Complex Equations

Author: Guo Chun Wen

Publisher: World Scientific

Published: 1999

Total Pages: 260

ISBN-13: 9789810238568

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"This is a very interesting book written by a well-known expert on complex methods in partial differential equations. It contains many recent results, many of them published for the first time, some published originally in Chinese".Mathematical Reviews

Nonlinear Parabolic and Elliptic Equations
Nonlinear Parabolic and Elliptic Equations

Author: C.V. Pao

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 786

ISBN-13: 1461530342

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In response to the growing use of reaction diffusion problems in many fields, this monograph gives a systematic treatment of a class of nonlinear parabolic and elliptic differential equations and their applications these problems. It is an important reference for mathematicians and engineers, as well as a practical text for graduate students.

Qualitative theory of parabolic equations. 1
Qualitative theory of parabolic equations. 1

Author: Tadeĭ Ivanovich Zeleni︠a︡k

Publisher: VSP

Published: 1997

Total Pages: 432

ISBN-13: 9789067642361

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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.

Fokker-Planck-Kolmogorov Equations
Fokker-Planck-Kolmogorov Equations

Author: Vladimir I. Bogachev

Publisher: American Mathematical Soc.

Published: 2015-12-17

Total Pages: 495

ISBN-13: 1470425580

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This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I
Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I

Author: Atsushi Yagi

Publisher: Springer Nature

Published: 2021-05-31

Total Pages: 68

ISBN-13: 9811618968

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The classical Łojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Łojasiewicz–Simon gradient inequality. This book presents a unified method to show asymptotic convergence of solutions to a stationary solution for abstract parabolic evolution equations of the gradient form by utilizing this Łojasiewicz–Simon gradient inequality. In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Łojasiewicz–Simon inequality is extended, which is published here for the first time. In the second version, these abstract results are applied to reaction–diffusion equations with discontinuous coefficients, reaction–diffusion systems, and epitaxial growth equations. The results are also applied to the famous chemotaxis model, i.e., the Keller–Segel equations even for higher-dimensional ones.

Nonlinear Evolution Equations And Their Applications - Proceedings Of The Luso-chinese Symposium
Nonlinear Evolution Equations And Their Applications - Proceedings Of The Luso-chinese Symposium

Author: Tatsien Li

Publisher: World Scientific

Published: 1999-08-31

Total Pages: 334

ISBN-13: 9814543446

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This book discusses recent trends and developments in the area of nonlinear evolution equations. It is a collection of invited lectures on the following topics: nonlinear parabolic equations (systems); nonlinear hyperbolic systems; free boundary problems; conservation laws and shock waves; travelling and solitary waves; regularity, stability and singularity, etc.