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Onset of solutions with internal layers approaching the domain boundary

Onset of solutions with internal layers approaching the domain boundary ISSN 0012-2661, Differential Equations, 2006, Vol. 42, No. 1, pp. 112–125.  c Pleiades Publishing, Inc., 2006. Original Russian Text  c I.V. Nedel’ko, 2006, published in Differentsial’nye Uravneniya, 2006, Vol. 42, No. 1, pp. 101–113. PARTIAL DIFFERENTIAL EQUATIONS Onset of Solutions with Internal Layers Approaching the Domain Boundary I. V. Nedel’ko Moscow State University, Moscow, Russia Received July 6, 2004 DOI: 10.1134/S0012266106010095 1. INTRODUCTION. STATEMENT OF THE PROBLEM Consider the problem ε (∆ u − u )= f (u, x, ε), (x, t) ∈ D × (0, +∞), (1) u(x, t, ε)= g(x), (x, t) ∈ ∂D × (0, +∞), (2) u(x, 0,ε)= u (x, ε),x ∈ D, (3) where ε> 0 is a small parameter, ∆ is the Laplace operator, x =(x ,x ), and D ⊂ R is a bounded 1 2 connected domain with sufficiently smooth boundary ∂D. Let the following condition be satisfied. ¯ ¯ (A1) There exist functions u ¯(x), u ˆ(x) ∈ C D such that u ¯(x) < u ˆ(x), x ∈ D, and the function f (u, x, 0) vanishes in the domain Π= (u, x): u¯(x) ≤ u ≤ u ˆ(x),x ∈ D only on the surfaces u = http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Onset of solutions with internal layers approaching the domain boundary

Nedel’ko, I.
Differential Equations , Volume 42 (1) – Mar 15, 2006
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References (5)

Publisher
Springer Journals
Copyright
Copyright © 2006 by Pleiades Publishing, Inc.
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266106010095
Publisher site
See Article on Publisher Site

Abstract

ISSN 0012-2661, Differential Equations, 2006, Vol. 42, No. 1, pp. 112–125.  c Pleiades Publishing, Inc., 2006. Original Russian Text  c I.V. Nedel’ko, 2006, published in Differentsial’nye Uravneniya, 2006, Vol. 42, No. 1, pp. 101–113. PARTIAL DIFFERENTIAL EQUATIONS Onset of Solutions with Internal Layers Approaching the Domain Boundary I. V. Nedel’ko Moscow State University, Moscow, Russia Received July 6, 2004 DOI: 10.1134/S0012266106010095 1. INTRODUCTION. STATEMENT OF THE PROBLEM Consider the problem ε (∆ u − u )= f (u, x, ε), (x, t) ∈ D × (0, +∞), (1) u(x, t, ε)= g(x), (x, t) ∈ ∂D × (0, +∞), (2) u(x, 0,ε)= u (x, ε),x ∈ D, (3) where ε> 0 is a small parameter, ∆ is the Laplace operator, x =(x ,x ), and D ⊂ R is a bounded 1 2 connected domain with sufficiently smooth boundary ∂D. Let the following condition be satisfied. ¯ ¯ (A1) There exist functions u ¯(x), u ˆ(x) ∈ C D such that u ¯(x) < u ˆ(x), x ∈ D, and the function f (u, x, 0) vanishes in the domain Π= (u, x): u¯(x) ≤ u ≤ u ˆ(x),x ∈ D only on the surfaces u =

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Differential Equations – Springer Journals

Published: Mar 15, 2006

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