Periodic traveling waves and locating oscillating patterns in multidimensional domains
Open Access
- 1 March 1999
- journal article
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 351 (7) , 2777-2805
- https://doi.org/10.1090/s0002-9947-99-02134-0
Abstract
We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R n \mathbb {R}^n , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.Keywords
This publication has 15 references indexed in Scilit:
- The Dirichlet Problem for Semilinear Second-Order Degenerate Elliptic Equations and Applications to Stochastic Exit Time Control ProblemsCommunications in Partial Differential Equations, 1995
- Generation and propagation of interfaces for reaction-diffusion equationsJournal of Differential Equations, 1992
- Eponential Decay To Stable States In Phase Transitions Via A Double Log–TransformationCommunications in Partial Differential Equations, 1990
- Behaviour of a semilinear periodic-parabolic problem when a parameter is smallPublished by Springer Nature ,1990
- The generation and propagation of internal layersNonlinear Analysis, 1988
- The Dynamics of Rotating Waves in Scalar Reaction Diffusion EquationsTransactions of the American Mathematical Society, 1988
- On the Singular Limit for a Cass of Problems Modelling Phase TransitionsSIAM Journal on Mathematical Analysis, 1987
- Stable transition layers in a semilinear boundary value problemJournal of Differential Equations, 1987
- A variational approach for a class of singular perturbation problems and applicationsProceedings of the Royal Society of Edinburgh: Section A Mathematics, 1987
- The approach of solutions of nonlinear diffusion equations to travelling wave solutionsBulletin of the American Mathematical Society, 1975