Article

Global Stability of Monostable Traveling Waves For Nonlocal Time-Delayed Reaction-Diffusion Equations

SIAM Journal on Mathematical Analysis (Impact Factor: 1.27). 01/2010; 42(6):2762-2790. DOI: 10.1137/090776342
Source: DBLP

ABSTRACT

For a class of nonlocal time-delayed reaction-diffusion equations, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near the negative infinity regardless of the magnitude of time delay. This work also improves and develops the existing stability results for local and nonlocal reaction-diffusion equations with delays. Our approach is based on the combination of the weighted energy method and the Green function technique.

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    • "Introduction. In this work, we study the existence and uniqueness of monotone wavefronts u(x, t) = φ(x+ct) for the monostable delayed non-local reaction–diffusion equation u t (t, x) = u xx (t, x) − u(t, x) + R K(x − y)g(u(t − h, y))dy, u ≥ 0, (1) when the reaction term g : R + → R + neither is monotone nor defines a quasi-monotone functional in the sense of Wu-Zou [47] or Martin-Smith [32] and when the non-negative kernel K(s) is Lebesgue integrable on R. Equation (1) is an important object of studies in the population dynamics, see [6] [14] [17] [20] [28] [29] [33] [34] [41] [42] [45] [48] [49] [50]. Taking formally K(s) = δ(s), the Dirac delta function, we obtain the diffusive Mackey-Glass type equation u t (t, x) = u xx (t, x) − u(t, x) + g(u(t − h, x)), u ≥ 0, (2) another popular focus of investigation, see [19] [24] [44] for more details and references. "
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    ABSTRACT: We propose a criterion for the existence of monotone wavefronts in non-monotone and non-local monostable diffusive equations of the Mackey-Glass type. This extends recent results by Gomez et al proved for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (up to a translation) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then this uniqueness actually holds within the class of all wavefronts and therefore the minimal fronts under consideration (either pulled or pushed) should be monotone. Similarly to the case of the KPP-Fisher equations, our approach is based on the construction of an appropriate fundamental solution for associated boundary value problem for linear integral-differential equation.
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    • "This problem is important because the spreading speeds c of the traveling waves in the biological applications usually are the minimum speed (i.e. the critical speed) [27] [36]. This problem is also challenging because the analytical approaches for stability of critical wavefronts by now are very limited, only case by case studies [18] [21]. Furthermore, when the critical traveling waves are oscillatory , their stability analysis is even more difficult. "
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    ABSTRACT: This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves exist for all and are exponentially stable for all wave speed [13], where is called the critical wave speed. In this paper, we prove that the critical traveling waves (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.
    Full-text · Article · Aug 2015 · Journal of Differential Equations
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    • "The nonlocal terms usually yield some gaps in the L 2 -energy estimates, which cause us to need to take c 1 so then we can control these gaps. In order to avoid such a trouble, for the case of nonlocal but still monotone equation, Mei-Ou-Zhao [26], Mei-Wang [27] and Huang-Mei- Wang [15] showed the stability for all (monotone) traveling waves with c ≥ c * by the L 1 -energy method but it sufficiently depends on the advantage of the monotonicity of both the equation itself and the traveling waves. Notice that, in this paper the equation (1) is nonlocal and non-monotone, and the traveling waves may be oscillating when r is big, so the above mentioned approaches, including the regular weighted energy method, the monotone method and Fourier transform method, they all seem to fail in obtaining the stability of the wavefronts for (1). "
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    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.
    Preview · Article · Aug 2015 · Discrete and Continuous Dynamical Systems
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