Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations. (English) Zbl 1515.35092
Summary: 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 \(g:\mathbb R_+\to\mathbb R_+\). 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.
Keywords:
monostable equation; reaction-diffusion equation; delay; super- and sub-solutions; wavefront; asymptotic stability; speed selectionReferences:
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