A class of pattern formation equations is treated, the existence of its global attractor is considered and some estirnates of its dimension are given. This sort of pattern formation equations relate to turbulence phenomenon in chemistry and combustion, so it is of great physical importance. Besides, its important theoretical value lies in that the equation contains differential operator of the fourth order in space. The global attractor exists after a series sophisticated estimates by the means of interpolation inequity and embedding theorems in sobolev spaces. Furthermore, some estimates on its fractal dimension are given.