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

This paper is concerned with the propagation phenomena for the nonlocal reaction-diffusion monostable equation with delay of the form partial derivative u(t,x)/partial derivative t = d Delta u(t, x) + f(u(t, x), integral(R) J(x - y)u(t - tau, y)dy), t > 0, x is an element of R. It is well-known that if we take J(x) = delta(x) and tau = 0, there exists a minimal wave speed c* > 0, such that this equation has no traveling wave front for 0 < c < c(*) and a traveling wave front for each c >= c(*), which is unique up to translation and is globally asymptotically stable. Furthermore, when J is symmetry and exponentially bounded and tau > 0, Wang et al. (2008) [27] considered the effects of delay and nonlocality on the spreading speed and proved that (i) if partial derivative(2)f(0,0)>0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if partial derivative(2)f(0,0)=0, then the delay and nonlocality do not affect the spreading speed. However, when J is asymmetry or exponentially unbounded, the question was left open. In this paper we obtain a rather complete answer to this question. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed. We also investigate the case of exponentially unbounded kernels where we prove the non-existence of traveling wave solutions and obtain upper and lower bounds for the position of any level set of the solutions. These bounds allow us to estimate how the solutions spread, depending on the kernels. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.