Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wave-fronts of the following equation: u(t)(x, t) = D[u(x + 1, t) + u(x - 1, t) - 2u(x, t)] - du(x, t) + b(u(x, t - r)), where x epsilon R, t > 0, D,d > 0, r >= 0, b epsilon C-1(R) and b(0)=dK - b(K)=0 for some K > 0 under monostable assumption. We show that there exists a minimal wave speed c* > 0, such that for each c > c* the equation has exactly one travelling wavefront U(x + ct) (up to a translation) satisfying U(-infinity) = 0, U(+infinity) = K and lim sup(xi ->-infinity) U(xi)e(-A1 (c)xi) < + infinity, where A(1)(c) is the smallest solution to the equation c lambda - D[e(lambda) + e(-lambda) - 2] + d - b'(0)e(-lambda cr) = 0. Moreover, the travelling wavefront is strictly monotone and asymptotically stable with phase shift in the sense that if an initial data phi epsilon C(R x [-r, 0] [0, K]) satisfies lim inf(x ->+infinity) phi(x, 0) > 0 and lim(x ->-infinity) max(s epsilon[-r,0])vertical bar phi(x, s)e(-A1 (c)x) - p(0)eA1((c)cs)vertical bar= 0 for some P-0 epsilon (0, +infinity), then the solution u(x, t) of the corresponding initial value problem satisfies lim(t ->+infinity) sup(R)vertical bar(-, t) /U( + ct + xi(0)) - 1 vertical bar = 0 for some xi(0) = xi(0)(U, phi) epsilon R. (c) 2005 Elsevier Inc. All rights reserved.