Existence and stability of solutions with periodically moving weak internal layers
- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 35B25 35B10 35K57 35K20
- singularly perturbed reaction diffusion equation, periodic boundary value, problem, boundary layers
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 $eprightarrow 0$.
- J. Math. Anal. Appl., 348 (2008) pp. 508-515.