Semi linear hyperbolic problems in bounded domains:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chur [u.a.]
Harwood Acad. Publ.
1987
|
| Series: | Mathematical reports
3,1 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 287 S. |
| ISBN: | 3718604604 |
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MARC
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| 100 | 1 | |a Haraux, Alain |e Verfasser |0 (DE-588)1166180271 |4 aut | |
| 245 | 1 | 0 | |a Semi linear hyperbolic problems in bounded domains |c Alain Haraux |
| 246 | 1 | 3 | |a Semi-linear hyperbolic problems in bounded domains |
| 264 | 1 | |a Chur [u.a.] |b Harwood Acad. Publ. |c 1987 | |
| 300 | |a XXIII, 287 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical reports |v 3,1 | |
| 830 | 0 | |a Mathematical reports |v 3,1 |w (DE-604)BV001892994 |9 3,1 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Editor s Introduction xi
Preface xiii
Notation xix
I. BACKGROUND 1
1.1. Basic Functional Setting 1
1.1.1. Maximal monotone linear operators 2
1.1.2. Abstract elliptic equations 4
1.1.3. Sobolev spaces 6
1.1.4. Elliptic equations of order *two . 10
1.2. Semi-Groups and the Abstract Wave
Equation 13
1.2.1. Vector-valued Sobolev spaces 14
1.2.2. Semi-groups and classical solutions of
inhomogeneous evolution equations 16
1.2.3. Classical solutions of Lipschitz
perturbations with forcing tefm 21
1.2.4. The abstract wave equation 26
1.3. Almost Periodicity and the Wave Equation 31
1.3.1. Basic facts on almost periodic functions 31
1.3.2. Precompact trajectories of the abstract
wave equation 34
1.3.3. An oscillation result 36
1.3.4. The wave equation in a bounded domain 39
V
vi CONTENTS
1.4. Some Technical Tools 40
1.4.1. Regularization techniques connected
with monotonicity and convexity 40
1.4.2. Properties of second-order elliptic
operators 43
1.4.3. Some differential and integral
inequalities 48
1.4.4. Additional facts from functional analysis 51
II. THE INITIAL VALUE PROBLEM 54
11.1. Strong Solutions of the Purely Dissipative
Equation 55
II. 1.1. An existence and uniqueness result 55
II. 1.2. Additional properties of strong
solutions 62
11.2. Weak Solutions 67
11.2.1. Weak solutions of equation (II.0.4) 68
11.2.2. Weak solutions of the full equation 75
11.2.3. Additional properties 80
11.3. Further Regularity Results 86
11.3.1. Regularity for equation (II.0.4) 87
11.3.2. Regular solutions of the full equation 90
11.4. Examples 95
11.4.1. Hyperbolic equations of order two in
space 95
11.4.2. Some higher-order examples 101
11.4.3. Remarks on the examples 104
III. ASYMPTOTICS IN SOME
AUTONOMOUS CASES 106
111.1. A Boundedness Property 106
111.2. Asymptotic Stability of the Equilibrium in
the Purely Dissipative Case 109
111.2.1. A general asymptotic stability property 109
111.2.2. Rate of decay to the equilibrium 118
CONTENTS vii
111.3. Decay to the Equilibrium when / is
Non-decreasing 126
111.3.1. Decay to zero when d=0 128
111.3.2. Convergence to the equilibrium when
h¥=0 133
111.3.3. Examples 133
111.4. Strong Dissipation and the Behavior of
Bounded Solutions 136
III.4.1. A general property 136
HI.4.2. Compactness criteria 139
III.4.3. Convergence to an equilibrium 140
111.5. More Difficult Situations 143
111.5.1. The case of a partial dissipation 143
111.5.2. More general dissipative terms 144
111.5.3. Consequences of stronger regularity 144
IV. NON-RESONANCE IN THE PURELY
DISSIPATIVE CASE 146
IV.l. The Resonance Phenomenon 146
IV.2. Bounded and Periodic Solutions 148
IV.2.1. A boundedness result 149
IV.2.2. Application to the existence of periodic
solutions 157
IV.3. Compactness and Almost Periodic
Solutions 160
IV.3.1. A compactness criterion 160
IV.3.2. More on almost-periodic functions 168
IV.3.3. An abstract existence theorem 172
IV.3.4. Application to equation (IV.2.0.1) 179
IV.4. Applications to the Basic Example 180
IV.4.1. Boundedness and periodic solutions 180
IV.4.2. Compactness and almost-periodic
solutions 182
IV.5. Additional Results and Perspectives of
Research 184
IV.5.1. On a result of G. Prodi 184
viii CONTENTS
IV.5.2. A weakened growth condition in two
dimensions 185
IV.5.3. More regular data in higher dimensions 185
IV.5.4. Main open problems 186
V. STABILITY OF PERIODIC AND ALMOST-
PERIODIC SOLUTIONS 187
V.I. The Set of Almost-Periodic Solutions 187
V.I.I. A uniqueness result on the linear
equation 188
V.1.2. The difference of two almost-periodic
solutions 190
V.I.3. A uniqueness result 192
V.2. Weak Convergence to an Almost-Periodic
Solution 193
V.2.1. Preliminary results 193
V.2.2. The main result 196
V.3. Strong Convergence Results 204
V.3.1. General results 205
V.3.2. Sufficient conditions for exponential
stability 206
V.3.3. Decay like a negative power of f 211
V.3.4. Comments on the rate of decay 213
V.4. Applications to the Main Example 215
V.4.1. General properties 215
V.4.2. The rate of decay in some typical cases 216
VI. OSCILLATION PROPERTIES IN THE
CONSERVATIVE CASE 218
VI.l. General Properties 218
VI. 1.1. Global solutions and the energy
conservation 218
VI. 1.2. Non-negative solutions in a cylinder 223
V1.2. The One-Dimensional Case 231
VI.2.1. A preliminary formula 232
CONTENTS ix
VI.2.2. A positivity result 233
VI.2.3. Proof of the theorem 234
VI.2.4. Comments on the behavior as |r| tends
to infinity 239
VI.3. The Linear Problem For n ^ 2 241
VI.3.1. Local oscillation properties of smooth
solutions 241
VI.3.2. A negative result in the rectangle 244
VI.3.3. Additional remarks on the linear case 245
VI.4. Extensions and Open Problems 246
VI.4.1. Spherically symmetric solutions of the
wave equation in a ball 246
VI.4.2. A semi-global oscillation result 247
VI.4.3. The vibrating string with a straight
obstacle 248
VI .4.4. Connection with questions of exact
controllability 249
VI.4.5. Main open problems 250
VII. GLOBAL PROPERTIES OF THE FULL
EQUATION 252
VII.1. Ultimate Boundedness When g is
Essentially Linear at Infinity 253
VI 1.2. Invariant Compact Sets and Attractors in
V x H 257
VII.2.1. Solutions bounded on U for a related
problem 257
VII.2.2. Convergence of the solutions to a
compact subset of V x H 262
VII.2.3. Construction of a compact attractor 263
VI 1.3. Application to the Semi-linear
Autonomous Wave Equation 267
VII.3.1. Application of the general theory 267
VII.3.2. Regularity of the attractor 269
VII.3.3. Additional properties 270
x CONTENTS
VII.4. Ultimate Boundedness in the General
Case 270
VII.5. Historical Comments and Perspectives of
Research 274
VII.5.1. The property of ultimate boundedness 274
VII.5.2. Attractors and fluid dynamics 275
VII.5.3. Open questions and possible
extensions 275
Bibliography 276
Author and Subject Indexes 283
|
| any_adam_object | 1 |
| author | Haraux, Alain |
| author_GND | (DE-588)1166180271 |
| author_facet | Haraux, Alain |
| author_role | aut |
| author_sort | Haraux, Alain |
| author_variant | a h ah |
| building | Verbundindex |
| bvnumber | BV026795795 |
| classification_rvk | SK 110 SK 540 SK 370 |
| ctrlnum | (OCoLC)917978904 (DE-599)BVBBV026795795 |
| discipline | Mathematik |
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| id | DE-604.BV026795795 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T23:19:10Z |
| institution | BVB |
| isbn | 3718604604 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022329086 |
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| owner_facet | DE-188 |
| physical | XXIII, 287 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Harwood Acad. Publ. |
| record_format | marc |
| series | Mathematical reports |
| series2 | Mathematical reports |
| spelling | Haraux, Alain Verfasser (DE-588)1166180271 aut Semi linear hyperbolic problems in bounded domains Alain Haraux Semi-linear hyperbolic problems in bounded domains Chur [u.a.] Harwood Acad. Publ. 1987 XXIII, 287 S. txt rdacontent n rdamedia nc rdacarrier Mathematical reports 3,1 Mathematical reports 3,1 (DE-604)BV001892994 3,1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022329086&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Haraux, Alain Semi linear hyperbolic problems in bounded domains Mathematical reports |
| title | Semi linear hyperbolic problems in bounded domains |
| title_alt | Semi-linear hyperbolic problems in bounded domains |
| title_auth | Semi linear hyperbolic problems in bounded domains |
| title_exact_search | Semi linear hyperbolic problems in bounded domains |
| title_full | Semi linear hyperbolic problems in bounded domains Alain Haraux |
| title_fullStr | Semi linear hyperbolic problems in bounded domains Alain Haraux |
| title_full_unstemmed | Semi linear hyperbolic problems in bounded domains Alain Haraux |
| title_short | Semi linear hyperbolic problems in bounded domains |
| title_sort | semi linear hyperbolic problems in bounded domains |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022329086&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001892994 |
| work_keys_str_mv | AT harauxalain semilinearhyperbolicproblemsinboundeddomains |