Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations

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