Reduction of weakly nonlinear parabolic partial differential equations

It is known that the Swift-Hohenberg equation ∂u/∂t=-(∂_x²+1)²+1)u+e{open}(u-u³) can be reduced to the Ginzburg-Landau equation (amplitude equation) ∂A/∂t=4∂_x²A+e{open}(A-3A|A|²) by means of the singular perturbation method. This means that if e{open} > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations ∂u/∂t=Pu+e{open}f(u) is proposed. n amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.