Doctoral School Mathematics and Scientific Computing

Abstract: Local well‑posedness for the generalized Camassa–Holm equation m_t + m_xu^p + bmu^p-1 u_x = -(g(u))_x + (b+1)u^p u_x, where p ≥ 1, k≥ 2, m = (I -∂_x²)^ku, g ∈ C^∞(ℝ, ℝ), and b ≠ 0 has already established for u₀ ∈ H^s, s>2(k-1)+3/2. We extend this to global well‑posedness under the constraints k ≥ 2, b=p+1, and m₀ ∈ L²(ℝ). Furthermore, we investigate propagation speeds, distinguishing regimes of finite vs. infinite speed. We show how the interplay between nonlinearity and dispersion determines whether compactly supported initial data remain compactly supported for all time or spread instantaneously. [hide abstract]