Gespeichert in:
| Titel: | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |
|---|---|
| Von: |
Peter Poláčik
|
| Person: |
Poláčik, Peter
1959- aut |
| Hauptverfasser: | |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
[2020]
|
| Schriftenreihe: | Memoirs of the American Mathematical Society
volume 264, number 1278 (first of 6 numbers) |
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032237705&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Zusammenfassung: | The author considers semilinear parabolic equations of the form u_t=u_xx+f(u),\quad x\in \mathbb R,t>0, where f a C 1 function. Assuming that 0 and \gamma >0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near \gamma for x\approx -\infty and near 0 for x\approx \infty . If the steady states 0 and \gamma are both stable, the main theorem shows that at large times, the graph of u(\cdot ,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(\cdot ,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, \gamma is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their \omega -limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories \{(u(x,t),u_x(x,t)):x\in \mathbb R\}, t>0, of the solutions in question. Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Main results -- 2.1. Minimal systems of waves and propagating terraces -- 2.2. The case where 0 and are both stable -- 2.3. The case where one of the steady states 0, is unstable -- 2.4. The \om-limit set and quasiconvergence -- 2.5. Locally uniform convergence to a specific front and exponential convergence -- Chapter 3. Phase plane analysis -- 3.1. Basic properties of the trajectories -- 3.2. A more detailed description of the minimal system of waves -- 3.3. Some trajectories out of the minimal system of waves -- Chapter 4. Proofs of Propositions 2.8, 2.12 -- Chapter 5. Preliminaries on the limit sets and zero number -- 5.1. Properties of ( ) -- 5.2. Zero number -- Chapter 6. Proofs of the main theorems -- 6.1. Some estimates: behavior at = and propagation -- 6.2. A key lemma: no intersection of spatial trajectories -- 6.3. The spatial trajectories of the functions in \Om( ) -- 6.4. \Om( ) contains the minimal propagating terrace -- 6.5. Ruling out other points from _{\Om}( ) -- 6.6. Completion of the proofs of Theorems 2.5, 2.13, and 2.15 -- 6.7. Completion of the proofs of Theorems 2.7, 2.9, 2.17 -- 6.8. Completion of the proofs of Theorems 2.11 and 2.19 -- 6.9. Proof of Theorem 2.22 -- Bibliography -- Back Cover. |
| Beschreibung: | Literaturverzeichnis: Seite 85-87 |
| Beschreibung: | v, 87 Seiten Diagramme, Illustrationen |
| ISBN: | 9781470441128 |
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| 245 | 1 | 0 | |a Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |c Peter Poláčik |
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| 490 | 1 | |a Memoirs of the American Mathematical Society |v volume 264, number 1278 (first of 6 numbers) | |
| 500 | |a Literaturverzeichnis: Seite 85-87 | ||
| 520 | |a The author considers semilinear parabolic equations of the form u_t=u_xx+f(u),\quad x\in \mathbb R,t>0, where f a C 1 function. Assuming that 0 and \gamma >0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near \gamma for x\approx -\infty and near 0 for x\approx \infty . If the steady states 0 and \gamma are both stable, the main theorem shows that at large times, the graph of u(\cdot ,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(\cdot ,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, \gamma is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their \omega -limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories \{(u(x,t),u_x(x,t)):x\in \mathbb R\}, t>0, of the solutions in question. | ||
| 520 | |a Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Main results -- 2.1. Minimal systems of waves and propagating terraces -- 2.2. The case where 0 and are both stable -- 2.3. The case where one of the steady states 0, is unstable -- 2.4. The \om-limit set and quasiconvergence -- 2.5. Locally uniform convergence to a specific front and exponential convergence -- Chapter 3. Phase plane analysis -- 3.1. Basic properties of the trajectories -- 3.2. A more detailed description of the minimal system of waves -- 3.3. Some trajectories out of the minimal system of waves -- Chapter 4. Proofs of Propositions 2.8, 2.12 -- Chapter 5. Preliminaries on the limit sets and zero number -- 5.1. Properties of ( ) -- 5.2. Zero number -- Chapter 6. Proofs of the main theorems -- 6.1. Some estimates: behavior at = and propagation -- 6.2. A key lemma: no intersection of spatial trajectories -- 6.3. The spatial trajectories of the functions in \Om( ) -- 6.4. \Om( ) contains the minimal propagating terrace -- 6.5. Ruling out other points from _{\Om}( ) -- 6.6. Completion of the proofs of Theorems 2.5, 2.13, and 2.15 -- 6.7. Completion of the proofs of Theorems 2.7, 2.9, 2.17 -- 6.8. Completion of the proofs of Theorems 2.11 and 2.19 -- 6.9. Proof of Theorem 2.22 -- Bibliography -- Back Cover. | ||
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Datensatz im Suchindex
| DE-BY-UBR_katkey | 6304352 |
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| _version_ | 1835110979629744128 |
| adam_text | American Mathematical Society
Number 1278
Propagating Terraces and the
Dynamics of Front-Like Solutions of
Reaction-Diffusion Equations on M
Peter Polacik
ULB Darmstadt
20396407
Universitäts- und
Landesbibüothek
Darmstadt
March 2020 • Volume 264 • Number 1278 (first of 6 numbers)
• ••a iin American
•:? A S Mathematical
society
Contents
Chapter 1 Introduction 1
Chapter 2 Main results 9
2 1 Minimal systems of waves and propagating terraces 10
2 2 The case where 0 and 7 are both stable 12
2 3 The case where one of the steady states 0, 7 is unstable 17
2 4 The w-limit set and quasiconvergence 20
2 5 Locally uniform convergence to a specific front and exponential
convergence 21
Chapter 3 Phase plane analysis 25
3 1 Basic properties of the trajectories 25
32A more detailed description of the minimal system of waves 32
3 3 Some trajectories out of the minimal system of waves 37
Chapter 4 Proofs of Propositions 2 8, 2 12 53
Chapter 5 Preliminaries on the limit sets and zero number 55
5 1 Properties of fi(u) 55
5 2 Zero number 55
Chapter 6 Proofs of the main theorems 59
6 1 Some estimates: behavior at x = ±00 and propagation 60
62A key lemma: no intersection of spatial trajectories 61
6 3 The spatial trajectories of the functions in U(u) 64
6 4 fi(u) contains the minimal propagating terrace 66
6 5 Ruling out other points from Ksi(u) 67
6 6 Completion of the proofs of Theorems 2 5, 2 13, and 2 15 72
6 7 Completion of the proofs of Theorems 2 7, 2 9, 2 17 74
6 8 Completion of the proofs of Theorems 2 11 and 2 19 78
6 9 Proof of Theorem 2 22 80
Bibliography 85
|
| any_adam_object | 1 |
| author | Poláčik, Peter 1959- |
| author_GND | (DE-588)12137906X |
| author_facet | Poláčik, Peter 1959- |
| author_role | aut |
| author_sort | Poláčik, Peter 1959- |
| author_variant | p p pp |
| building | Verbundindex |
| bvnumber | BV046828499 |
| classification_rvk | SI 130 |
| ctrlnum | (OCoLC)1193288257 (DE-599)BVBBV046828499 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV046828499 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T19:02:09Z |
| institution | BVB |
| isbn | 9781470441128 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032237705 |
| oclc_num | 1193288257 |
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| owner | DE-29T DE-355 DE-BY-UBR DE-83 DE-11 |
| owner_facet | DE-29T DE-355 DE-BY-UBR DE-83 DE-11 |
| physical | v, 87 Seiten Diagramme, Illustrationen |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spellingShingle | Poláčik, Peter 1959- Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R Memoirs of the American Mathematical Society Mittelwertsatz Integralrechnung (DE-588)4332255-4 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Anfangswertproblem (DE-588)4001991-3 gnd |
| subject_GND | (DE-588)4332255-4 (DE-588)4173245-5 (DE-588)4001991-3 |
| title | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |
| title_auth | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |
| title_exact_search | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |
| title_full | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R Peter Poláčik |
| title_fullStr | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R Peter Poláčik |
| title_full_unstemmed | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R Peter Poláčik |
| title_short | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R |
| title_sort | propagating terraces and the dynamics of front like solutions of reaction diffusion equations on r |
| topic | Mittelwertsatz Integralrechnung (DE-588)4332255-4 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Anfangswertproblem (DE-588)4001991-3 gnd |
| topic_facet | Mittelwertsatz Integralrechnung Parabolische Differentialgleichung Anfangswertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032237705&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT polacikpeter propagatingterracesandthedynamicsoffrontlikesolutionsofreactiondiffusionequationsonr |
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