Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains

J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 1, Page 109--179.

Mimura, Masayasu ; Sakamoto, Kunimochi
Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]


Abstract:
By using singular perturbation techniques, it is shown that an exothermal reaction-diffusion system with a small parameter in long cylindrical domains admits a family of transition layer solutions. The solutions exhibit spatial inhomogeneity in two directions, one in the axis of the cylinder and the other in the cross-section of the cylindrical domain. The profile of the solutions in the cross-sectional direction is determined by a family of solutions of a non-linear elliptic eigenvalue problem, called {\it the perturbed Gelfand problem}. On the other hand, the profile of the solutions in the axial direction of the cylindrical domain has a sharp transition layer. The stability analysis is also carried out for the equilibrium solutions, which reveals that a Hopf-bifurcation occurs as some control parameters are varied, exhibiting spatio-temporal oscillations.

Mathematics Subject Classification (1991): Primary 35B25; Secondary 35B4, 35K57
Mathematical Reviews Number: MR1414623

Received: 1995-03-09