Stability of semi-wavefronts for delayed reaction-diffusion equations

Abstract

This paper deals with the asymptotic behavior of solutions to the delayed monostable equation: $(\ast )$ $u_t(t,x)=u_{xx}(t,x)-u(t,x)+g(u(t-h,x)),$ $x\in \mathbb{R},t>0,$ 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 $leadingedge$ of semi-wavefronts for $(\ast )$ with $g$ satisfying $g(u)\le g^{\prime }(0)u,u\in {\mathbb{R}}_{+},$ is established