On the cauchy problem for certain system of semilinear parabolic equations

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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ \begin{gathered} \psi _t = - (\sigma - \alpha )\psi - \sigma \theta _x + \alpha \psi _{xx} , \hfill \\ \theta _t = - (1 - \beta )\theta + v\psi _x + (\psi \theta )_x + \beta \theta _{xx} . \hfill \\ \end{gathered} \right.$$ \end{document}