Stability of planar travelling waves for bistable reaction–diffusion equations in multiple dimensions

Abstract

This paper proves the multidimensional stability with an algebraic rate as time goes to infinity of planar travelling waves for bistable reaction–diffusion equations in for all , without assuming the hypothesis on the spectral gap of the one-dimensional linearized operator in as in previous works. The convergence rate in obtained here is better than the known rate in . Compared with the stability results where the convergence rate is only given for the special initial data which are the translations of the travelling wave, the convergence rate obtained in the present paper is for much more general initial perturbations.

Keywords:

• multidimensional stability
• travelling waves
• bistable reaction–diffusion equations

AMS Subject Classifications:

• 35K55
• 35K57
• 35Q92