Reduction of weakly nonlinear parabolic partial differential equations

TL;DR

This paper extends the reduction technique from the Swift-Hohenberg equation to a broader class of weakly nonlinear parabolic PDEs, providing amplitude equations, error estimates, and stability results for solutions.

Contribution

It introduces a systematic reduction method for nonlinear parabolic PDE systems, including error bounds and stability analysis, generalizing previous specific cases.

Findings

Amplitude equations approximate original PDE solutions when b5 is small.

Stable steady states in amplitude equations imply stable periodic solutions in original PDEs.

Error estimates hold uniformly near periodic solutions.

Abstract

It is known that the Swift-Hohenberg equation $\partial u/\partial t = -(\partial_x^2 + 1)^2u + \varepsilon (u-u^3)$ can be reduced to the Ginzburg-Landau equation (amplitude equation) $\partial A/\partial t = 4\partial_x^2 A + \varepsilon (A-3A|A|^2)$ by means of the singular perturbation method. This means that if $\varepsilon >0$ 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 $\partial u/\partial t = \mathcal{P}u + \varepsilon f(u)$ is proposed. An 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 $t>0$.