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

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Abstract

We study the asymptotic stability of traveling fronts and the front’s velocity selection problem for the time-delayed monostable equation \((*)\) \(u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\) \(x \in {\mathbb R},\ t >0,\) with Lipschitz continuous reaction term \(g: {\mathbb R}_+ \rightarrow {\mathbb R}_+\). We also assume that g is \(C^{1,\alpha }\)-smooth in some neighbourhood of the equilibria 0 and \(\kappa >0\) to \((*)\). In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of g so that equation \((*)\) can possess the pushed traveling fronts. Our first main result says that the non-critical wavefronts of \((*)\) with monotone g are globally nonlinearly stable. In the special and easier case when the Lipschitz constant for g coincides with \(g'(0)\), we prove a series of results concerning the exponential (asymptotic) stability of non-critical (respectively, critical) fronts for the monostable model \((*)\). As an application, we present a criterion of the absolute global stability of non-critical wavefronts to the diffusive non-monotone Nicholson’s blowflies equation.

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Acknowledgments

The authors express their gratitude to the anonymous referee whose valuable comments helped to improve the original version of the paper. This research was supported by FONDECYT (Chile) 1150480. We also thank Viktor Tkachenko (Institute of Mathematics in Kyiv, Ukraine) and Robert Hakl (Mathematical Institute in Brno, Czech Republic) for useful discussions. Especially we would like to acknowledge FONDECYT (Chile), project 1110309, and CONICYT (Chile), Project MEC 80130046, for supporting the research stays of Dr. Tkachenko and Dr. Hakl, respectively, at the University of Talca.

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  1. Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

    Abraham Solar & Sergei Trofimchuk

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  1. Abraham Solar
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Correspondence to Sergei Trofimchuk.

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Solar, A., Trofimchuk, S. Speed Selection and Stability of Wavefronts for Delayed Monostable Reaction-Diffusion Equations. J Dyn Diff Equat 28, 1265–1292 (2016). https://doi.org/10.1007/s10884-015-9482-6

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  • DOI: https://doi.org/10.1007/s10884-015-9482-6

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