Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations

We study the existence and asymptotic stability of traveling waves to u(t) = [g(u(+1)) + g(u(-1)) - 2g(u)] + f(u) on R x (0,infinity), where u = u(x, t), u+/-1 = u(x 1, t), g = du(P) (d > 0, p greater than or equal to 1) and f = u - u(2). We show that there exists (c) over bar > 0 such that for each wave speed e >, there is a traveling wave U e Cl (R), i.e., a solution of the form u = U(x - ct). The traveling wave has the property that U(-infinity) = 1, U'<0 on R, and limxi-->infinity. U(xi)e(lambdaxi) = 1, where lambda = Lambda(1)(c) is the smallest solution to clambda = f(1)(0) + g(1)(0)[e + e(-lambda)) - 2]. We also show that the traveling wave is globally asymptotically stable in the sense that if an initial value u((.), 0) is an element of C(R --> [0,1]) satisfies lim inf(x-->-infinity) u(x, 0) > 0 and lim(x-->infinity) u(x,0)e(lambdax) = 1 for some lambda is an element of (0, Lambda(1) ((c) over bar)), then lim(t-->infinity) sup(R) \u((.) + ct, t)/U((.)) - 1\ = 0 where (c, U) is the traveling wave with speed c = C(lambda) = {f'(0) + g'(0)[e(lambda) + e(-lambda) - 2]}/lambda, the inverse of lambda = Lambda(1)(c). (C) 2002 Elsevier Science (USA).