Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

Abstract

This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossingmonostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson's blowflies equation in population dynamics and Mackey-Glass model in physiology. (2000). 35K57 · 35R10 · 35B40 · 92D25. Mathematics Subject Classification

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

“…Notice that these stabilities rely on the monotonicity of both the equation and the wavefronts. However, when p d > e, equation (1.1) lacks monotonicity and the traveling waves may oscillate around v + ; research on the stability of such oscillatory waves was only recently carried out in [40]. Because of the lack of monotonicity, the equation doesn't possess the comparison principle, and we cannot expect global stability.…”

“…The choice of η and the weight function w(ξ) are also delicate and important, because with a different setting for η and w(ξ), we may not be able to get the positivity (3.11) of A η,w (ξ) for every speed c > c * . Otherwise, as in [28,27,18,39,40], we need to restrict the speed to be large, c c * . It is worth pointing out that such a technical selection of η = e λcr was first given by Gourley in [9] on the linear stability of wavefronts for an age-structured population model.…”

Speed Selection and Stability of Wavefronts for Delayed Monostable Reaction-Diffusion Equations

“…Another important approach to the wave stability problem in (1) is a weighted energy method developed by Mei et al [26,27,28,29]. See also Kyrychko et al [16], Lv and Wang [19], Wu et al [47]. This method is based on rather technical weighted energy estimations and generally requires better properties from g and w 0 .…”

“…In the last decade, the wavefront solutions of equation (11) have been investigated by many authors, e.g. see [6,7,14,18,19,20,26,28,29,45,47]. If the positive parameters p, δ are such that 1 < p/δ ≤ e, then g is monotone and satisfies the hypothesis (H) with L g = g ′ (0) and κ = ln(p/δ).…”

Traveling Wavefronts on Reaction Diffusion Systems with Spatio-Temporal Delays

“…Traveling wave solutions, usually characterized as solutions invariant with respect to translation in space, have attracted much attention due to their significant nature in science and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In which, the theory of wave fronts of reaction diffusion systems is an important part, and its history traces back to the so-called Fisher-KPP equation, the celebrated mathematical works by P. A. Fisher and by Kolmogorov, Petrovskii and Piscunov.…”