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 $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, 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 $(*)$. 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 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 present 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 Nicholson's blowflies equation.