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In this article, we give a variational formulation for traveling wave solutions that decay exponentially at one end of the cylinder for parabolic equations. The variational formulation allows us to obtain the monotone dependence of the velocity on the domain and the nonlinearity, since the velocity is related to the infimum. In particular, we apply this method to Ginzburg-Landau-type problems and a scalar reaction-diffusion-advection equation in infinite cylinders. For the former, we not only obtain the existence, non-existence, boundedness and regularity of the solutions, but also obtain the monotone dependence of the velocity on the nonlinearity and the domain. For the later, we obtain the monotone dependence of the velocity on the nonlinearity and the domain besides the existence, uniqueness, monotonicity and asymptotic behavior at infinity of the solutions. Moreover, we deduce that the influence of the advection on the traveling waves is different from a flow along the cylinder axis considered in many articles.

In the previous article [Y.-X. Wang and W.-T. Li, J. Differential Equations, 251 (2011) 1670-1695], the authors have shown that the set of positive stationary solutions of a cross-diffusive Lotka-Volterra cooperative system can form an unbounded fish-hook shaped branch $Gamma_p$. In the present paper, we will show some criteria for the stability of positive stationary solutions on $Gamma_p$. Our results assert that if $d_1/d_2$ is small enough, then unstable positive stationary solutions bifurcate from semitrivial solutions, the stability changes only at every turning point of $Gamma_p$ and no Hopf bifurcation occurs. While as $d_1/d_2$ becomes large, the stability has a drastic change when $mu