Properties of the solutions to the two-component -family systems

Global well-posedness and blow-up for the damped perturbation of Camassa-Holm type equations

“…, where λ > 0 is the dissipative parameter, κ = ±1, and the real dimensionless constant b ∈ R is a parameter that provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching; it is also the number of covariant dimensions associated with the momentum density m. The unknown u(t, x) stands for the horizontal velocity of the fluid, and σ (t, x) is related to the free surface elevation from equilibrium with the boundary assumption, u → 0 and σ → 1 as |x| → ∞. The system (1.1) without the dissipative terms has been studied by several authors, such as [6,7,15,26,27]. Note that the system (1.1) can be reduced to the damping perturbation of some well-known Camassa-Holm type equations.…”

On blow‐up phenomena for a weakly dissipative periodic two‐component b$$ b $$‐family system revisited

“…Wang [3] proved that the solution map of the Cauchy problem of (1.1) was not uniformly continuous in the periodic condition. Zong [4] studied the existence of global solutions, persistence properties, and propagation speed for the system (1.1). The global well-posedness of (1.1) in the space H s−1,p (R) × H s,p (R), s > max…”

“…Wang [3] proved that the solution map of the Cauchy problem of () was not uniformly continuous in the periodic condition. Zong [4] studied the existence of global solutions, persistence properties, and propagation speed for the system (). The global well‐posedness of () in the space $$$ {H}^{s-1,p}\left(\mathrm{\mathbb{R}}\right)\times {H}^{s,p}\left(\mathrm{\mathbb{R}}\right),s>\max \left\{2,\frac{3}{2}+\frac{1}{p}\right\},p\in \left(1,\infty \right) $$$ by Du and Wu in [5].…”

Non-uniform continuity on initial data for the two-component b-family system in Besov space

“…The local well-posedness space was further enlarged, and established in Besov space B s p,r × B s−1 p,r with s > max{1 + 1 p , 3 2 }, 1 ≤ p, r ≤ ∞ (however, for r = ∞, the continuity of the data-to-solution map is established in a weaker topology) [29,23]. Some aspects concerning blow-up scenario, global solutions, persistence properties and propagation speed, see the discussions in [24,30].…”