We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ 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 analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.

中文翻译：

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膜方程的全局近平面对称解

我们证明任何空间维膜方程的简单平面行波解 $d\geqslant 3$ 具有有限空间范围的在足够小的紧凑支持的扰动下是全局非线性稳定的，其中小幅度取决于扰动支持的大小以及初始行波轮廓。该论点的主要新颖之处在于我们的基于矢量场的方法中缺乏高阶剥离。特别是，高阶能量（事实上，所有有序的能量 $2$ 或更高）被允许以多项式（但以受控方式）及时增长。这与经典的全局稳定性论证形成对比，其中只有自举论证中使用的“最高”阶能量表现出增长，并反映了背景行波解具有“无限能量”且微扰方程的系数不是这一事实渐近洛伦兹不变量。尽管如此，我们可以证明微扰在 $C^{2}$ 通过仔细分析非线性相互作用并揭示方程中的某个“残留”零结构。