000			04474nam a22004453i 4500
001			EBC6195961
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005			20240724114250.0
006			m o d |
007			cr cnu||||||||
008			240724s2020 xx o ||||0 eng d
020			_a9781470458065 _q(electronic bk.)
020			_z9781470441128
035			_a(MiAaPQ)EBC6195961
035			_a(Au-PeEL)EBL6195961
035			_a(OCoLC)1154528262
040			_aMiAaPQ _beng _erda _epn _cMiAaPQ _dMiAaPQ
050		4	_aQA377 _b.P653 2020
082	0		_a515/.3534
100	1		_aPoláčik, Peter.
245	1	0	_aPropagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on _ Mathbb{R} _.
250			_a1st ed.
264		1	_aProvidence : _bAmerican Mathematical Society, _c2020.
264		4	_c©2020.
300			_a1 online resource (100 pages)
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
490	1		_aMemoirs of the American Mathematical Society Series ; _vv.264
505	0		_aCover -- 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 Ω( ) -- 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.
520			_aThe 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.
588			_aDescription based on publisher supplied metadata and other sources.
590			_aElectronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
650		0	_aPartial differential equations -- Qualitative properties of solutions -- Stability.
655		4	_aElectronic books.
776	0	8	_iPrint version: _aPoláčik, Peter _tPropagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on _ Mathbb{R} _dProvidence : American Mathematical Society,c2020 _z9781470441128
797	2		_aProQuest (Firm)
830		0	_aMemoirs of the American Mathematical Society Series
856	4	0	_uhttps://ebookcentral.proquest.com/lib/orpp/detail.action?docID=6195961 _zClick to View
999			_c18132 _d18132