Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
Boca Raton, Fla. [u.a.]
Chapman & Hall/CRC
2014
|
| Series: | Monographs and research notes in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXVI, 543 S. graph. Darst. |
| ISBN: | 9781482251722 |
Staff View
MARC
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| 100 | 1 | |a Galaktionov, Victor A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations |c Victor A. Galaktionov ; Enzo L. Mitidieri ; Stanislav I. Pohozaev |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b Chapman & Hall/CRC |c 2014 | |
| 300 | |a XXVI, 543 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Monographs and research notes in mathematics | |
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| 700 | 1 | |a Mitidieri, Enzo |d 1954- |e Verfasser |0 (DE-588)1045645842 |4 aut | |
| 700 | 1 | |a Pochožaev, Stanislav I. |d 1935- |e Verfasser |0 (DE-588)129018880 |4 aut | |
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Record in the Search Index
| _version_ | 1804152647223083008 |
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| adam_text | Contents
fì
Introduction:
Self-Similar Singularity Patterns for Various Higher-
Order Nonlinear Partial Differential Equations
xiii
First four ID examples: basics of a unified approach
.........xiii
Layout of Chapter
1:
four nonlinear PDEs, blow-up, other patterns
xvi
Layout of Chapter
2:
non-blow-up for
semilinear
parabolic PDEs
. xix
Layout of Chapter
3: semilinear
parabolic Kuramoto-Sivashmsky,
Navier-Stokes, and Burnett models
...............xix
Layout of Chapter
4:
blow-up for
a
quasilinear
parabolic PDE
. . . xx
Layout of Chapter
5:
blow-up patterns for
a
semilinear
fourth-order
hyperbolic PDE
..........................xxi
Layout of Chapter
6:
a
quasilinear
fourth-order hyperbolic PDE
. . xxii
Layout of Chapter
7:
Korteweg—
de Vries
equations
..........xxiii
Layout of Chapter
8:
shocks for nonlinear dispersion PDEs
.....xxiii
Layout of Chapter
9:
higher-order
Schrödinger
equations
......xxv
1
Complicated Self-Similar Blow-up, Compacton, and Standing
Wave Patterns for Four Nonlinear PDEs: a Unified
Variational Approach to Elliptic Equations
1
1.1
Introduction: higher-order evolution models, self-similar blow¬
up,
compactons,
and standing wave solutions
......... 2
1.2
Problem blow-up : parabolic and hyperbolic PDEs
... 10
1.3
Problem existence : variational approach to countable
families of solutions by the Lusternik— Schnirel man category
and Pohozaev s fibering theory
................. 16
1.4
Problem oscillations : local oscillatory structure of solu¬
tions close to interfaces
...................... 29
1.5
Problem numerics : a first classification of basic types of
localized blow-up or compacton patterns for
m
— 2......37
1.6
Problem numerics : patterns for
m
> 3.......... 48
1.7
Toward smoother PDEs: fast diffusion
............. 52
1.8
New families of patterns: Cartesian fibering
.......... 57
1.9
Problem Sturm index : a homotopy classification of pat¬
terns via e-reguiarization
..................... 63
1.10
Problem fast diffusion : extinction and blow-up phe¬
nomenon in the Dirichlet setting
................ 74
1.11
Problem fast diffusion : L-S and other patterns
..... 79
ix
χ
Contents
1.12 Non-L-S
patterns: linearized algebraic approach
...... 88
1.13
Problem Sturm index :
Ä-compression..........97
1.14 Quasilinear
extensions: a gradient diffusivity
......... 99
2
Classification of Global Sign-Changing Solutions of
Semilinear
Heat Equations in the
Subcriticai
Fujita Range: Second- and
Higher-Order Diffusion
103
2.1 Semilinear
heat PDEs, blow-up, and global solutions
.....103
2.2
Countable set of p-branches of global self-similar solutions:
general strategy
..........................107
2.3
Pitchfork /^bifurcations of profiles
...............109
2.4
Global p-bifurcation branches: fibering
.............115
2.5
Countable family of global linearized patterns
.........120
2.6
Some structural properties of the set of global solutions via
critical points: blow-up, transversality, and connecting orbits
123
2.7
On evolution completeness of global patterns
.........127
2.8
Higher-order PDEs: non-variational similarity and center sub-
space patterns
...........................129
2.9
Global similarity profiles and bifurcation branches
......130
2.10
Numerics: extension of even p-branches of profiles
.......140
2.11
Odd non-symmetric profiles and their p-branches
.......149
2.12
Second countable family: global linearized patterns
......154
3
Global and Blow-up Solutions for Kuramoto—Sivashinsky,
Navier—
Stokes, and Burnett Equations
157
3.1
Introduction: Kuramoto-Sivashinsky, Navier-Stokes, and Bur¬
nett equations
...........................157
3.2
Interpolation: global existence for the
К
SE ..........
163
3.3
Method of eigenfunctions: blow-up
...............165
3.4
Global existence by weighted Gronwall s inequalities
.....168
3.5
Global existence and L°°-bounds by scaling techniques
.... 172
3.6
L°°-bounds for the Navier-Stokes equations in RN and well-
posed Burnett equations
.....................182
4
Regional, Single-Point, and Global Blow-up for a Fourth-
Order Porous Medium-Type Equation with Source
189
4.1 Semilinear
and
quasilinear
blow-up reaction-diffusion models
189
4.2
Fundamental solution and spectral properties:
η
— 0 .....196
4.3
Local properties of solutions near interfaces
..........197
4.4
Blow-up similarity solutions
...................201
4.5
Regional blow-up profiles for
ρ
=
η
4- 1.............204
4.6
Single-point blow-up for p>n+l
...............218
4.7
Global blow-up profiles for
ρ
є
(1,
η
+ 1)............228
Contents xi
5 Semilinear
Fourth-Order Hyperbolic Equation:
Tuvo
Types
of Blow-up Patterns
237
5.1
Introduction:
semilinear
wave equations and blow-up patterns
237
5.2
Fundamental solution of the linear PDE and local existence
. 243
5.3
Rescaled equation and related Hermitian spectral theory
. . . 246
5.4
Construction of linearized blow-up patterns
..........256
5.5
Self-similar blow-up: nonlinear eigenfunctions
.........260
6 Quasilinear
Fourth-Order Hyperbolic Boussinesq Equation:
Shock, Rarefaction, and Fundamental Solutions
271
6.1
Introduction:
quasilinear
Boussinesq (wave) model and shocks
271
6.2
Shock formation blow-up similarity solutions
.........276
6.3
Fundamental solution as a nonlinear eigenfunction
......284
7
Blow-up and Global Solutions for
Korteweg—de Vries-
Type
equat ions
291
7.1
Introduction: KdV equation and blow-up
...........291
7-2
Method of investigation: blow-up via nonlinear capacity
. . . 297
7.3
Proofs of blow-up results
.....................302
7.4
The Cauchy problem for the KdV equation
..........303
8
Higher-Order Nonlinear Dispersion PDEs: Shock, Rarefac¬
tion, and Blow-up Waves
309
8.1
Introduction: nonlinear dispersion PDEs and main problems
. 309
8.2
First blow-up results by two methods
..............318
8.3
Shock and rarefaction waves for Szp(x),
fř(±)(or),
etc
.....322
8.4
Unbounded shocks and other singularities
...........329
8.5
TWs and generic formation of moving shocks
.........339
8.6
The Cauchy problem for NDEs: smooth deformations,
com¬
pactons,
and extensions to higher orders
............343
8.7
Conservation laws: smooth ¿-deformations
...........345
8.8
On ¿-entropy solutions (a test) of the NDE
..........348
8.9
On extensions to other related NDEs
..............359
8.10
On related higher-order in time NDEs
.............369
8.11
On shocks for spatially higher-order NDEs
...........375
8.12
Changing sign
compactons
for higher-order NDEs
.......378
8.13
NDE-3: gradient blow-up and nonuniqueness
.........382
8.14
Gradient blow-up similarity solutions
..............391
8.15
Nonunique extensions beyond blow-up
.............396
8.16
NDE-3: parabolic approximation
................409
8.17
Fifth-order NDEs and main problems
..............415
8.18
Problem blow-up : shock S_ solutions
..........419
8.19
Riemann problems S±: rarefactions and shocks
......429
8.20
Nonuniqueness after shock formation
.............437
8.21
Shocks for NDEs with the Cauchy-Kovalevskaya theorem
. . 453
xii Contents
8.22 Problem
oscillatory
compactons
for fifth- and seventh-
order NDEs
............................456
9
Higher-Order
Schrödinger
Equations: from Blow-up Zero
Structures to
Quasilinear
Operators
461
9.1
Introduction: duality of global and blow-up scalings, Her-
mitian spectral theory, and refined scattering
.........461
9.2
The fundamental solution and the convolution
.........466
9.3
Discrete real spectrum and eigenfunctions of
В
........468
9.4
Spectrum and polynomial eigenfunctions of B*
........478
9.5
Application I: evolution completeness of
Φ
in L2p. {]RN), sharp
estimates in ]R++1, extensions
.................484
9.6
Applications II and III: local structure of nodal sets and
unique continuation by blow-up scaling
.............489
9.7
Application IV: a boundary point regularity via a blow-up
micro-analysis
...........................492
9.8
Application V: toward countable families of nonlinear eigen¬
functions of the QLSE
......................503
9.9
Extras: eigenfunction expansions and little Hubert spaces
. .510
References
515
Index
537
List of Frequently Used Abbreviations
543
Mathematics
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and
Schrödinger
Equations shows how four types of higher-order nonlinear
evolution partial differential equations (PDEs) have many commonalities
through their special
quasilinear
degenerate representations. The authors
present a unified approach to deal with these
quasilinear
PDEs.
The book first studies the particular self-similar singularity solutions (pat¬
terns) of the equations. This approach allows four different classes of
nonlinear PDEs to be treated simultaneously to establish their striking
common features. The book describes many properties of the equations
and examines traditionai questions of existence/nonexistence, unique-
ness/nonuniqueness. global asymptotics. regularizations, shock-wave
theory, and various blow-up singularities.
Features
•
Shows how complex PDEs are used in various applications
•
Presents a unified variational approach to analyze four types
of nonlinear equations: parabolic, hyperbolic, dispersion, and
Schrödinger
•
Describes new nonlinear phenomena for the equations
•
Enables readers to prove blow-up for each type of PDE using
nonlinear capacity and generalized eigenfunction methods
•
Uses homotopy and branching approaches to deal with
nonvariational elliptic problems
•
Explains the formation of discontinuous shock waves for a broad
class of nonlinear dispersion equations from compacton theory
Preparing readers for more advanced mathematical PDE analysis, this
book demonstrates that
quasilinear
degenerate higher-order PDEs. even
exotic and awkward ones, are not as daunting as they first appear. It also
illustrates the deep features shared by several types of nonlinear PDEs
and encourages readers to further develop this unifying PDE approach
from other viewpoints.
|
| any_adam_object | 1 |
| author | Galaktionov, Victor A. Mitidieri, Enzo 1954- Pochožaev, Stanislav I. 1935- |
| author_GND | (DE-588)1045645842 (DE-588)129018880 |
| author_facet | Galaktionov, Victor A. Mitidieri, Enzo 1954- Pochožaev, Stanislav I. 1935- |
| author_role | aut aut aut |
| author_sort | Galaktionov, Victor A. |
| author_variant | v a g va vag e m em s i p si sip |
| building | Verbundindex |
| bvnumber | BV042153725 |
| classification_rvk | SK 540 SK 560 |
| ctrlnum | (OCoLC)905017785 (DE-599)OBVAC11866334 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042153725 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:14:02Z |
| institution | BVB |
| isbn | 9781482251722 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027593498 |
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| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XXVI, 543 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series2 | Monographs and research notes in mathematics |
| spelling | Galaktionov, Victor A. Verfasser aut Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations Victor A. Galaktionov ; Enzo L. Mitidieri ; Stanislav I. Pohozaev Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2014 XXVI, 543 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Monographs and research notes in mathematics Schrödinger-Gleichung (DE-588)4053332-3 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s Hyperbolische Differentialgleichung (DE-588)4131213-2 s Schrödinger-Gleichung (DE-588)4053332-3 s DE-604 Mitidieri, Enzo 1954- Verfasser (DE-588)1045645842 aut Pochožaev, Stanislav I. 1935- Verfasser (DE-588)129018880 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027593498&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027593498&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Galaktionov, Victor A. Mitidieri, Enzo 1954- Pochožaev, Stanislav I. 1935- Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations Schrödinger-Gleichung (DE-588)4053332-3 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd |
| subject_GND | (DE-588)4053332-3 (DE-588)4173245-5 (DE-588)4131213-2 |
| title | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations |
| title_auth | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations |
| title_exact_search | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations |
| title_full | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations Victor A. Galaktionov ; Enzo L. Mitidieri ; Stanislav I. Pohozaev |
| title_fullStr | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations Victor A. Galaktionov ; Enzo L. Mitidieri ; Stanislav I. Pohozaev |
| title_full_unstemmed | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations Victor A. Galaktionov ; Enzo L. Mitidieri ; Stanislav I. Pohozaev |
| title_short | Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations |
| title_sort | blow up for higher order parabolic hyperbolic dispersion and schrodinger equations |
| topic | Schrödinger-Gleichung (DE-588)4053332-3 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd |
| topic_facet | Schrödinger-Gleichung Parabolische Differentialgleichung Hyperbolische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027593498&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027593498&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT galaktionovvictora blowupforhigherorderparabolichyperbolicdispersionandschrodingerequations AT mitidierienzo blowupforhigherorderparabolichyperbolicdispersionandschrodingerequations AT pochozaevstanislavi blowupforhigherorderparabolichyperbolicdispersionandschrodingerequations |