# Global nearly-plane-symmetric solutions to the membrane equation

## TL;DR

This paper proves the nonlinear stability of certain planar traveling wave solutions to the membrane equation in higher dimensions, allowing polynomial growth of energies and revealing a vestigial null structure.

## Contribution

It introduces a novel vector-field method that handles infinite energy solutions without higher-order peeling, enabling stability analysis with polynomial energy growth.

## Key findings

- Global nonlinear stability of planar traveling waves in dimension d ≥ 3.
- Allowing polynomial growth of higher-order energies in stability analysis.
- Identification of a vestigial null structure in the membrane equation.

## Abstract

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d \geq 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly-supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher-order peeling in our vector-field based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments where only the "top" order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has "infinite energy" and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^2$ by carefully analyzing the nonlinear interactions and exposing a certain "vestigial" null structure in the equations.