Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter

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Abstract

Semilinear elliptic equations of the form ?ni,j=1(aij(cursive Greek chi)ucursive Greek chii(cursive Greek chi))cursive Greek chij + f(?, cursive Greek chi, u(cursive Greek chi)) = 0, cursive ? O, u(cursive Greek chi) - 0, cursive Greek chi ? ?O, are considered, where ? ? R is a parameter, O ? Rn is a bounded domain and f is a smooth non-linear function. It is shown that for 'generic' functions f, the set of non-trivial solutions (?, u) consists of a finite, or countable, collection of smooth, 1-dimensional curves and any such solution is either hyperbolic or is a saddle-node bifurcation point of the curve.

Original languageEnglish
Pages (from-to)730-739
Number of pages10
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume47
Issue number5
Publication statusPublished - Sep 1996

Keywords

  • Generic
  • Hyperbolic
  • Reaction-diffusion equations
  • Saddle-node bifurcation
  • Stationary solutions

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