traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity

∂u/∂t= j ∗ u − u + f (u) on r × (0,∞),

where the dispersion kernel j is nonnegative and the nonlinearity f is monostable type. we show that there exists c^∗ ∈ r such that for any c > c∗, there is a nonincreasing traveling wave solution φ with φ(−∞) = 1 and lim_ξ→∞φ(ξ )e^λξ = 1, where λ = λ1(c) is the smallest positive solution to cλ =∫_r j(z)e^λzdz −1+f ′(0). furthermore, the existence of traveling wave solutions with c = c^∗ is also established. for c ̸= 0, we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense.