Stability theorem for singularly perturbed solutions to systems of reaction-diffusion equations
type: Item
instance of: scholarly article
title: Stability theorem for singularly perturbed solutions to systems of reaction-diffusion equations
zbMATH Open document ID: 0626.35009
author: Yasumasa Nishiura
author: Hiroshi Fujii
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: The author presents a stability theorem to singularly perturbed stationary solutions (SPS) of the systems of nonlinear diffusion equations with a small parameter \(\epsilon >0:\) \[ u_ t=\epsilon^ 2u_{xx}+f(u,v)\quad and\quad v_ t=Dv_{xx}+g(u,v),\quad x\in I=(0,1),\quad t>0,\quad u_ x=0=v_ x,\quad x\in \partial I=\{0,1\}. \] A new idea to solve the stability problem of SPS of one mode type for a general D is given. The singular limit eigenvalue problem: \[ \epsilon^ 2w_{xx}+f_ u^{\epsilon} w+f_ v^{\epsilon} z=\lambda w\quad and\quad Dz_{xx}+g_ u^{\epsilon} w+g_ v^{\epsilon} z=z,\quad in\quad I,\quad w_ x=0=z_ x\quad on\quad \partial I \] plays a key role. Authors' main conclusion is: Under suitable assumptions, there exists only one critical eigenvalue \(\lambda =\lambda_ 0(\epsilon)\) which is real and simple. When \(\epsilon\downarrow 0\), it behaves like \(\lambda_ 0(\epsilon)\simeq -\nu \epsilon\) \((\nu >0)\).
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 4017437
zbMATH Keywords: stability
zbMATH Keywords: singularly perturbed stationary solutions
zbMATH Keywords: nonlinear diffusion
zbMATH Keywords: small parameter
zbMATH Keywords: one mode type
zbMATH Keywords: singular limit eigenvalue problem
MaRDI profile type: Publication
full work available at URL: pjaa.61.329
OpenAlex ID: W1984581442
cites work: A density-dependent diffusion system with stable discontinuous stationary solutions
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cites work: Boundary and interior transition layer phenomena for pairs of second- order differential equations
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DOI: 10.3792/PJAA.61.329
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description: scientific article; zbMATH DE number 4017437