Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations

In this work we consider the global asymptotic stability of pushed traveling fronts for one-dimensional monostable reaction-diffusion equations with monotone delayed reactions. Pushed traveling front is a special type of critical wave front which converges to zero more rapidly than the near non- critical wave fronts. Recently, Trofimchuk et al. [16] proved the existence and uniqueness of pushed traveling fronts of the considered equation when the re-action term lost the sub-tangency condition. In this article, using the comparison method via a pair of super- and sub-solution and squeezing technique, we prove that the pushed traveling fronts are globally exponentially stable. This also gives an affirmative answer to an open problem presented by Solar and Trofimchuk [14].