Stability of planar traveling fronts in bistable reaction-diffusion systems

This paper is concerned with the multidimensional stability of planar traveling fronts in bistable reaction-diffusion systems. It is first shown that planar traveling fronts are asymptotically stable under spatially decaying initial perturbations by appealing to the comparison principle and super-subsolution method. In particular, if the perturbations belong to L1(Rn−1) in a certain sense, we obtain a convergence rate like t−n−12. Then we show that the solution of the Cauchy problem converges to the planar traveling front with rate t−n+14 for a spatially non-decaying perturbation with the help of semigroup theory. Finally, we prove that there exists a solution oscillating permanently between two planar traveling fronts, which indicates that planar traveling fronts are not always asymptotically stable in multidimensional space under general bounded perturbations.