Global nearly-plane-symmetric solutions to 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.