Abstract
This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.
Highlights
To our previous study [15] on the stability of monotone traveling waves to the nonlocal dispersion equation, in this paper we further2010 Mathematics Subject Classification
The main target of the present paper is to show the stability of the monotone/nonmonotone wavefronts to (1) for all c > c∗, where the wave speed c under consideration can be allowed sufficiently close to the minimum wave speed c∗
For the local Nicholson’s blowflies equation, Lin-LinLin-Mei [19] succeeded in obtaining the stability of all monotone/non-monotone traveling waves with c > c∗, by means of the regular L2-weighted energy method with a new development by a nonlinear Halanay’s inequality
Summary
To our previous study [15] on the stability of monotone traveling waves to the nonlocal dispersion equation, in this paper we further2010 Mathematics Subject Classification. Consider the stability of non-monotone traveling waves to the nonlocal dispersion equation with time-delay vt − D(J ∗ v − v) + d(v) = K ∗ b(v(t − r, ·)), (1)
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