Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations

The existence and uniqueness are proved for global classical solutions of the spatially periodic Cauchy problem to the following system of parabolic equations ctsy s ya .c ysuxqacx x u sy 1yb u qnc q cu qbu , . .x t x x x which was proposed as a substitute for the Rayleigh]Benard equation and can lead to Lorenz equations.