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.

We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.(1.9) with z decreasing from z 0 to 0 and with Z increasing from 0 to Z ∞ . To better understand the crucial relationship between the asymptotic phaseshift Z ∞ and the additive perturbation z, we note that the super-solution residual J = u t − u xx − g(u) is given by(1.10)Close to the interface, the term g(Φ) − g(Φ + z) ∼ −g ′ (Φ)z is negative and must be dominated by the positive termŻΦ ′ . This requires thatŻ dominate z andż. On the other hand, close to the Post-interaction regime -convergence to waveContinuous Setting In this final step of the program, the large transients generated in the interaction regime must be controlled in a frame that moves along with the unobstructed wave. As discussed above, this analysis leads naturally to a large basin nonlinear stability result for planar travelling wave solutions to the unobstructed PDE (1.2) with Ω = R 2 . For presentation purposes, we will focus our discussion here on this unobstructed special case. Indeed, the inclusion of the obstacle merely adds technical complications that do not contribute to the understanding of the differences between the continuous and discrete frameworks. 7The main task is to construct a super-solution for (1.2) with Ω = R 2 of the form( 1.15) which adds transverse effects to the Ansatz (1.9) discussed earlier. As before, the function z decreases from z 0 to 0 while Z increases from 0 to Z ∞ . Both z and Z should be thought of as small terms. By contrast, the new function θ should be allowed to be arbitrarily large at t = 0, provided that it is localized in the sense θ(·, 0) ∈ L 2 and that it decays to zero as t → ∞ uniformly in y. This function controls deformations of the wave interface in the transverse direction. We note that any localized initial perturbation from the wave can be dominated by the initial condition in (1.15) by choosing z 0 positive and as small as we wish, at the cost of a larger value for θ(·, 0) L 2 . Assuming for the moment that Z ∞ scales with z 0 , this freedom implies that we can dominate the transients caused by such a perturbation by a family of super-solutions that have arbitrarily small asymptotic phase offsets Z ∞ . A similar argument with sub-solutions then establishes the convergence to the planar wave without any asymptotic phase shift.The difference in behaviour between one and two spatial dimensions is hence caused by the extra transverse direction, al...

No abstract

We show that the fast travelling pulses of the discrete FitzHugh-Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be verified involve linear operators that are associated to functional differential equations of mixed type. Such equations are ill-posed and do not admit a semi-flow, which precludes the use of standard Evans-function techniques. Instead, we construct the potential eigenfunctions directly by using exponential dichotomies, Fredholm techniques and an infinitedimensional version of the Exchange Lemma.AMS 2010 Subject Classification: 34A33, 34K26, 34D35, 34K08.