Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity

This paper is concerned with traveling wave solutions of the equation ∂u∂t=J∗u−u+f(u)onR×(0,∞), where the dispersion kernel JJ is nonnegative and the nonlinearity ff is monostable type. We show that there exists c∗∈Rc∗∈R such that for any c>c∗c>c∗, there is a nonincreasing traveling wave solution ϕϕ with ϕ(−∞)=1ϕ(−∞)=1 and limξ→∞ϕ(ξ)eλξ=1, where λ=Λ1(c)λ=Λ1(c) is the smallest positive solution to cλ=∫RJ(z)eλzdz−1+f′(0). Furthermore, the existence of traveling wave solutions with c=c∗c=c∗ is also established. For c≠0c≠0, we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense.