In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:\documentclass[12pt]{minimal}
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$$\left\{ \begin{gathered} \psi _t = - (\sigma - \alpha )\psi - \sigma \theta _x + \alpha \psi _{xx} , \hfill \\ \theta _t = - (1 - \beta )\theta + v\psi _x + (\psi \theta )_x + \beta \theta _{xx} . \hfill \\ \end{gathered} \right.$$
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