Random traveling wave and bifurcations of asymptotic behaviors in the stochastic KPP equation driven by dual noises

In this paper, we study the influences of dually environmental noises on the traveling wave which develops from the deterministic KPP equation. We prove that if the strengths of noises satisfy some condition, the solution of the stochastic KPP equation with Heaviside initial condition develops a random traveling wave, whose wave speed is deterministic and depends on the strengths of noises. If the strengths of noises satisfy some other conditions, the solution tends to zero as time tends to infinity. Therefore, there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises.