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Author: Atsushi Yagi
Publisher: Springer Science & Business Media
ISBN: 3642046312
Category : Mathematics
Languages : en
Pages : 594
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Book Description
This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0
Author: Atsushi Yagi
Publisher: Springer Science & Business Media
ISBN: 3642046312
Category : Mathematics
Languages : en
Pages : 594
Get Book Here
Book Description
This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0
Author: Atsushi Yagi
Publisher: Springer
ISBN: 9789811618956
Category : Mathematics
Languages : en
Pages : 61
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Book Description
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.
Author: D Daners
Publisher: Chapman and Hall/CRC
ISBN:
Category : Mathematics
Languages : en
Pages : 268
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Book Description
Part of the Pitman Research Notes in Mathematics series, this text covers: linear evolution equations of parabolic type; semilinear evolution equations of parabolic type; evolution equations and positivity; semilinear periodic evolution equations; and applications.
Author: Irena Lasiecka
Publisher: Cambridge University Press
ISBN: 9780521434089
Category : Mathematics
Languages : en
Pages : 678
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Book Description
Originally published in 2000, this is the first volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which is unifying across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 1 includes the abstract parabolic theory for the finite and infinite cases and corresponding PDE illustrations as well as various abstract hyperbolic settings in the finite case. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.
Author: Alessandra Lunardi
Publisher: Springer Science & Business Media
ISBN: 3034805578
Category : Mathematics
Languages : en
Pages : 437
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Book Description
The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones. Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived. The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques. - - - This very interesting book provides a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and how this theory may be used in the study of parabolic partial differential equations; it takes into account the developments of the theory during the last fifteen years. (...) For instance, optimal regularity results are a typical feature of abstract parabolic equations; they are comprehensively studied in this book, and yield new and old regularity results for parabolic partial differential equations and systems. (Mathematical Reviews) Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hölder continuous derivatives. (Zentralblatt MATH)
Author: G Lumer
Publisher: CRC Press
ISBN: 9780824790103
Category : Medical
Languages : en
Pages : 534
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Book Description
This volume presents a collection of lectures on linear partial differntial equations and semigroups, nonlinear equations, stochastic evolutionary processes, and evolution problems from physics, engineering and mathematical biology. The contributions come from the 6th International Conference on Evolution Equations and Their Applications in Physical and Life Sciences, held in Bad Herrenalb, Germany.
Author: Gisele Ruiz Goldstein
Publisher: CRC Press
ISBN: 9780824709754
Category : Mathematics
Languages : en
Pages : 442
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Book Description
Celebrating the work of renowned mathematician Jerome A. Goldstein, this reference compiles original research on the theory and application of evolution equations to stochastics, physics, engineering, biology, and finance. The text explores a wide range of topics in linear and nonlinear semigroup theory, operator theory, functional analysis, and linear and nonlinear partial differential equations, and studies the latest theoretical developments and uses of evolution equations in a variety of disciplines. Providing nearly 500 references, the book contains discussions by renowned mathematicians such as H. Brezis, G. Da Prato, N.E. Gretskij, I. Lasiecka, Peter Lax, M. M. Rao, and R. Triggiani.
Author: Herbert Amann
Publisher: Birkhäuser
ISBN: 3034892217
Category : Mathematics
Languages : en
Pages : 366
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Book Description
In this treatise we present the semigroup approach to quasilinear evolution equa of parabolic type that has been developed over the last ten years, approxi tions mately. It emphasizes the dynamic viewpoint and is sufficiently general and flexible to encompass a great variety of concrete systems of partial differential equations occurring in science, some of those being of rather 'nonstandard' type. In partic ular, to date it is the only general method that applies to noncoercive systems. Although we are interested in nonlinear problems, our method is based on the theory of linear holomorphic semigroups. This distinguishes it from the theory of nonlinear contraction semigroups whose basis is a nonlinear version of the Hille Yosida theorem: the Crandall-Liggett theorem. The latter theory is well-known and well-documented in the literature. Even though it is a powerful technique having found many applications, it is limited in its scope by the fact that, in concrete applications, it is closely tied to the maximum principle. Thus the theory of nonlinear contraction semigroups does not apply to systems, in general, since they do not allow for a maximum principle. For these reasons we do not include that theory.
Author: F. Kappel
Publisher: Pitman Publishing
ISBN:
Category : Mathematics
Languages : en
Pages : 352
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Book Description
Author: Tomás Roubicek
Publisher: Springer Science & Business Media
ISBN: 3764373970
Category : Mathematics
Languages : en
Pages : 415
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Book Description
This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. The exposition quickly leads general theory to analysis of concrete equations, which have specific applications in such areas as electrically (semi-) conductive media, modeling of biological systems, and mechanical engineering. Methods of Galerkin or of Rothe are exposed in a large generality.
