发表机构
Sichuan Normal University; Fudan University（四川师范大学; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究广义Boussinesq方程孤立波附近的全局动力学，证明偶-奇扰动在流形外要么散射要么有限时间爆破，通过不变区域分类和一次通过定理实现二分法。
AI 中文摘要
我们研究了广义Boussinesq方程在驻立孤立波附近的全局动力学。对于偶-奇扰动，Maulen [J. Math. Pures Appl. 177 (2023)] 在孤立波Q附近构造了一个渐近稳定的中心-稳定流形M。我们通过证明Q的任何充分小的偶-奇扰动若位于M之外，则要么在能量空间中当t趋于正无穷时散射，要么在有限时间内爆破，从而完善了这一局部描述。证明依赖于两个主要要素。首先，我们通过识别分别与散射和爆破相关的两个不变区域，对能量低于基态能量的偶-奇解进行分类。其次，我们建立了一维中的一次通过定理：非散射解一旦离开孤立波的适当选取邻域，就不能重新进入该邻域。因此，解被限制在两个不变区域之一中。结合上述分类，这给出了M之外散射与爆破之间的二分法。
英文摘要
We study the global dynamics of the generalized Boussinesq equation near standing solitary waves. For even-odd perturbations, Maulen [J. Math. Pures Appl. 177 (2023)] constructed an asymptotically stable center-stable manifold M near the solitary wave Q. We complete this local description by proving that any sufficiently small even-odd perturbation of Q lying outside M either scatters in the energy space as t -> +infinity or blows up in finite time. The proof relies on two main ingredients. First, we classify even-odd solutions with energy below the ground-state energy by identifying two invariant regions, associated with scattering and blow-up, respectively. Second, we establish a one-pass theorem in one dimension: a non-scattering solution cannot re-enter a suitably chosen neighborhood of the solitary wave once it has left that neighborhood. Consequently, the solution remains confined to one of the two invariant regions. Together with the classification above, this yields a dichotomy between scattering and blow-up outside M.
