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On the pulse dynamics for reaction-diffusion systems on one-dimensional domains with various boundary conditions

島谷, 晴基北海道大学

2023.03.23

概要

Linearized eigenvalue problems are problems to investigate the stability of stationary
solutions for reaction-diﬀusion systems. Analytical methods for eigenvalues and eigenfunctions have been studied intensively for a long time. Sturm-Liouville theory and the SLEP
method ([31], [32]), which were developed in the course of these studies, are still powerful
tools for linearized eigenvalue problems. However, there are still few general methods to
analyze linearized eigenvalue problems. Against this background, many researchers have
studied this problem. In the process, studies of the one-dimensional Allen-Cahn equation
with the Neumann boundary condition have been developed.
(
∂t u = ϵ2 ∂xx u + u(1 − u2 ), t > 0, x ∈ (0, 1),
(3.1.1)
∂x u(t, 0) = ∂x u(t, 1) = 0,
where 0 < ϵ ≪ 1. (3.1.1) was rigorously studied by [4]. Fix r ∈ N. By many studies, we
see that there exists a stationary

solution u(x; ϵ) of (3.1.1) satisfying being suﬃciently
Pr
x−zj
j
close to j=1 (−1) tanh √2ϵ (j = 1, . . . , r), where zj = 2j−1
. Here r ∈ N denote the
2r
number of a stable front-type stationary solution S(x) of (3.1.1) and u(x; ϵ) is a function of
x with a parameter ϵ. Furthermore, starting from (3.1.1), various researches for 3.1.1 was
performed(e.g. [1], [2], [14]). Thereafter, Wakasa-Yotsutani([35], [36], [37], [38]) analyzed
next linearized eigenvalue problem for u(x; ϵ):
(
ϵ2 ∂xx φ(x) + F ′ (u(x; ϵ))φ(x) + λϕ(x) = 0, x ∈ (0, 1),
(3.1.2)
∂x φ(0) = ∂x φ(1) = 0,
where 0 < ϵ ≪ 1 and F (u) = u(1 − u2 ) or F (u) = sin u. [35], [36], [37], [38] reported
that all eigenvalues λn (n = 0, 1, 2, 3, . . .) of (3.1.2) and eigenfunctions φn (x) associated
with λn hold the following. In 3.1, ∼ means asymptotic equivalence, and we denote
r,ϵ
λn = λr,ϵ
n , φn = φn in the following (1) and (2). In addtion, The following results (1) and
(2) are rigorously proved in [35], [36], [37], [38]. ...

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...

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On the pulse dynamics for reaction-diffusion systems on one-dimensional domains with various boundary conditions

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On the pulse dynamics for reaction-diffusion systems on one-dimensional domains with various boundary conditions

概要

この論文で使われている画像

関連論文

A WELL-POSEDNESS FOR THE REACTION DIFFUSION EQUATIONS OF BELOUSOV-ZHABOTINSKY REACTION

On uniqueness for Schrödinger maps with low regularity large data

Restoration of chiral symmetry in cold and dense Nambu-Jona-Lasinio model with tensor renormalization group

STOCHASTIC HYPERBOLIC SYSTEMS, SMALL PERTURBATIONS AND PATHWISE APPROXIMATION

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