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The Resource Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, by Marcelo R. Ebert, Michael Reissig, (electronic resource)

Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, by Marcelo R. Ebert, Michael Reissig, (electronic resource)

Label
Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models
Title
Methods for Partial Differential Equations
Title remainder
Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models
Statement of responsibility
by Marcelo R. Ebert, Michael Reissig
Creator
Contributor
Author
Subject
Language
eng
Summary
This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area. The book is organized in five parts: In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation. Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes Laplace equation, heat equation and wave equation as well as the different features of each theory. It also discusses the notion of energy of solutions, a highly effective tool for the treatment of non-stationary or evolution models and shows how to define energies for different models. Part 3 demonstrates how phase space analysis and interpolation techniques are used to prove decay estimates for solutions on and away from the conjugate line. It also examines how terms of lower order (mass or dissipation) or additional regularity of the data may influence expected results. Part 4 addresses semilinear models with power type non-linearity of source and absorbing type in order to determine critical exponents: two well-known critical exponents, the Fujita exponent and the Strauss exponent come into play. Depending on concrete models these critical exponents divide the range of admissible powers in classes which make it possible to prove quite different qualitative properties of solutions, for example, the stability of the zero solution or blow-up behavior of local (in time) solutions. The last part features selected research projects and general background material.--
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Assigning source
Provided by publisher
http://library.link/vocab/creatorName
Ebert, Marcelo R
Image bit depth
0
Literary form
non fiction
Nature of contents
dictionaries
http://library.link/vocab/relatedWorkOrContributorName
Reissig, Michael
Series statement
Springer eBooks
http://library.link/vocab/subjectName
  • Mathematics
  • Partial differential equations
Label
Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, by Marcelo R. Ebert, Michael Reissig, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=http://dx.doi.org/10.1007/978-3-319-66456-9
Instantiates
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Part 1 -- Introduction -- Part 2 -- Partial differential equations in models -- Basics for partial differential equations -- The Cauchy-Kovalevskaja theorem -- Holmgren’s uniqueness theorem -- Method of characteristics -- Burger’s equation -- Laplace equation - properties of solutions - starting point of elliptic theory -- Heat equation - properties of solutions - starting point of parabolic theory -- Wave equation - properties of solutions - starting point of hyperbolic theory -- Energies of solutions - one of the most important quantities -- Part 3 -- Phase space analysis for heat equation -- Phase space analysis and smoothing for Schrödinger equations -- Phase space analysis for wave models -- Phase space analysis for plate models -- The method of stationary phase and applications -- Part 4 -- Semilinear heat models -- Semilinear classical damped wave models -- Semilinear wave models with a special structural dissipation -- Semilinear classical wave models -- Semilinear Schrödinger models -- Linear hyperbolic systems -- Part 5 -- Research projects for beginners -- Background material
Control code
978-3-319-66456-9
Dimensions
unknown
Extent
1 online resource (XVI, 456 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783319664569
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-66456-9
Other physical details
1 illustration
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1026408145
Label
Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, by Marcelo R. Ebert, Michael Reissig, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=http://dx.doi.org/10.1007/978-3-319-66456-9
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Part 1 -- Introduction -- Part 2 -- Partial differential equations in models -- Basics for partial differential equations -- The Cauchy-Kovalevskaja theorem -- Holmgren’s uniqueness theorem -- Method of characteristics -- Burger’s equation -- Laplace equation - properties of solutions - starting point of elliptic theory -- Heat equation - properties of solutions - starting point of parabolic theory -- Wave equation - properties of solutions - starting point of hyperbolic theory -- Energies of solutions - one of the most important quantities -- Part 3 -- Phase space analysis for heat equation -- Phase space analysis and smoothing for Schrödinger equations -- Phase space analysis for wave models -- Phase space analysis for plate models -- The method of stationary phase and applications -- Part 4 -- Semilinear heat models -- Semilinear classical damped wave models -- Semilinear wave models with a special structural dissipation -- Semilinear classical wave models -- Semilinear Schrödinger models -- Linear hyperbolic systems -- Part 5 -- Research projects for beginners -- Background material
Control code
978-3-319-66456-9
Dimensions
unknown
Extent
1 online resource (XVI, 456 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783319664569
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-66456-9
Other physical details
1 illustration
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1026408145

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