Stability of semi-wavefronts for delayed reaction-diffusion equations

This paper deals with the asymptotic behavior of solutions to the delayed monostable equation: $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in \mathbb{R},\ t >0,$ where $h>0$ and the reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$ has exactly two fixed points (zero and $\kappa >0$). Under certain condition on the derivative of $g$ at $\kappa$, the global stability of fast wavefronts is proved. Also, the stability of the $leading \ edge$ of semi-wavefronts for $(*)$ with $g$ satisfying $g(u)\leq g'(0)u, u\in\R_+,$ is established