Linear and superlinear spreading speeds of monostable equations with nonlocal delayed effects (Q6615816)

In this paper, the authors study the propagation phenomena for the following nonlocal reaction-diffusion monostable equation with delay\N\[\N\frac{\partial u(t,x)}{\partial t}=d\Delta u(t,x)+f\left(u(t,x), \int_\mathbb{R}J(x-y)u(t-\tau, y)dy\right).\N\]\NWhen \(J\) is symmetric and exponentially bounded and \(\tau\ge 0\), \textit{Z.-C. Wang} et al. established the existence, nonexistence, uniqueness and stability of traveling wave solutions [J. Dyn. Differ. Equations 20, No. 3, 573--607 (2008; Zbl 1141.35058)]. However, when \(J\) is asymmetric or exponentially unbounded, the question was left open. In this paper, the authors are devoted to giving a rather complete answer to this question.\N\NBy applying the abstract monotone dynamical systems theory developed in [\textit{X. Liang} and \textit{X.-Q. Zhao}, Commun. Pure Appl. Math. 60, No. 1, 1--40 (2007; Zbl 1106.76008)], the authors first show that for exponentially bounded but asymmetric kernels, the minimal speed of traveling waves exists and coincides with the spreading speed. They also obtain the effects of delay and nonlocality on the spreading speed. Furthermore, they consider the acceleration phenomena for exponentially unbounded kernels. More precisely, they investigate the case of algebraically decaying kernels, and prove the non-existence of traveling wave solutions, and show the level sets of the solutions eventually locate in between two exponential functions of time.