Stability of Traveling Waves for Reaction-Diffusion Equations with Multiplicative Noise

Abstract

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small.By applying a stochastic phase-shift together with a time-transform, we obtain a semilinear SPDE that describes the fluctuations from the primary wave. We subsequently develop a semigroup approach to handle the nonlinear stability question in a fashion that is closely related to modern deterministic methods.

Travelling waves for reaction–diffusion equations forced by translation invariant noise

“…We shall see that (5.17) can be used to recover the results in [16,17] from the expressions that we derive in this section.…”

“…On account of the estimates in Appendix A, the bounds in [17, §7] can be transferred to the current context. The result can hence be established by following the proof of [17,Prop. 2.2].…”

“…The right hand side of this equation is now completely orthogonal to ψ tw for small V . 5 In the case of 1d Brownian motion, we explain in [17] how g can be chosen to make this possible.…”

“…As in [17], the techniques developed in [35,Chapter 3] can be used to treat the second component of (5.40) as an SDE for Γ with random coefficients. The key ingredient is [35, Thm.…”

“…holds for any w ∈ W and ψ ∈ L 2 (R n ). This point of view allows us to unify the framework of this paper with the setup used in [16,17] where scalar noise is considered. Indeed, for the setting described in §1-4 we can take W = L 2 (R m ) and W Q = L 2 Q .…”