Remarks on Large-Scale Effects of Smoothing Mechanisms in 3D Reaction-Diffusion Equations

D. Erhard, Weijun Xu

2021, v.27, Issue 4, 505-521

ABSTRACT

In \cite{Phi4_general_smoothing}, we considered a class of reaction-diffusion equations that approximates the dynamical $\Phi^4_3$ model at large scales. These approximations involve very general smoothing mechanisms that are higher order perturbations of the Laplacian. In this note, we discuss assumptions made on the smoothing mechanism in that article. In particular, we remark that the strict positivity condition of the Fourier multiplier of the operator cannot be relaxed without other modifications, not even to allow it to reach zero outside the origin. On the other hand, if we introduce suitable Fourier cutoffs that are compatible with the non-positive part of the operator, then we expect that no smoothing assumption will be needed on its higher order terms. We then explain how this changes the coupling constant of the limiting equation, and how to modify the argument to prove the result.

Keywords: large-scale effects, general smoothing mechanism, reaction-diffusion equations

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