Article

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

(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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Available from: Ming Mei,
• "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. "
##### Article: Monotone waves for non-monotone and non-local monostable reaction-diffusion equations
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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. "
##### Article: Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay
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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.
Journal of Differential Equations 08/2015; 259(4). DOI:10.1016/j.jde.2015.03.003 · 1.68 Impact Factor
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• "It is natural to expect that the rate of convergence in (8) is exponential, see e.g. [10] [27] [28] [33] [35]. The demonstration of this fact, however, is based on a different approach and will be considered in a separate work. "
##### Article: Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction
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ABSTRACT: We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on $R$, every solution of equation $(*)$ with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sharp form of the theory of spreading speeds for equation $(*)$.
Nonlinearity 08/2014; 28(7). DOI:10.1088/0951-7715/28/7/2027 · 1.21 Impact Factor