WIAS Preprint No. 1294, (2008)

Existence and stability of solutions with periodically moving weak internal layers



Authors

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

2010 Mathematics Subject Classification

  • 35B25 35B10 35K57 35K20

Keywords

  • singularly perturbed reaction diffusion equation, periodic boundary value, problem, boundary layers

Abstract

We consider the periodic parabolic differential equation $ep^2 Big( fracpartial^2 upartial x^2 -fracpartial upartial t Big)=f(u,x,t,ep)$ under the assumption that $ve$ is a small positive parameter and that the degenerate equation $f(u,x,t,0) =0$ has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution $u_p(x,t,ep)$ which is $T$-periodic in $t$, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as $eprightarrow 0$.

Appeared in

  • J. Math. Anal. Appl., 348 (2008) pp. 508-515.

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