WIAS Preprint No. 646, (2001)

On a class of singularly perturbed partly dissipative reaction-diffusion systems



Authors

  • Butuzov, Valentin F.
  • Nefedov, Nikolai N.
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 35B25 35K57

Keywords

  • Initial boundary value problem, singularly perturbed partly dissipative reaction-diffusion system, exchange of stabilities, asymptotic lower and upper solutions

Abstract

We consider the singularly perturbed partly dissipative reaction-diffusion system $ve^2 left(fracpartial upartial t - fracpartial^2 upartial x^2right) =g(u,v,x,t,ve), fracpartial vpartial t = f(u,v,x,t,ve)$ under the condition that the degenerate equation $g(u,v,t,0)=0$ has two solutions $u= varphi_i(v,x,t), i=1,2,$ that intersect (exchange of stabilities) and that $v$ is a vector. Our main result concerns existence and asymptotic behavior in $ve$ of the solution of the initial boundary value problem under consideration. The proof is based on the method of asymptotic lower and upper solutions.

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