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We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations, 1 $$\left\{ \begin{aligned} & \psi _{t} = - (1 - \alpha )\psi - \theta _{x} + \psi \psi _{x} + \alpha \psi _{xx} , & \theta _t = - (1 - \alpha )\theta + \nu \psi _{x} + (\psi \theta )_{x} + \alpha \theta _{xx}\end{aligned} \right.$$ . for initial data with different end states, 2 $$(\psi (x,0),\theta (x,0)) = (\psi _0 (x),\theta _{0} (x)) \to (\psi _{ \pm} ,\theta _ {\pm} ),\quad {\rm as}\quad x \to {\pm}\infty,\quad(2)$$ which displays the complexity in between ellipticity and dissipation. Due to smoothing effect of the parabolic operator, we detail the regularity property and estimates when t > 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data. Also we discuss the decay estimates without the restriction of L 1 bound as in Tang and Zhao [17], Wang [20]. Related to recent work by [15], our derivation may also establish the same estimates directly if under the same condition.