Abstract
In this paper, we investigate the modified two-component Camassa-Holm equation with kappa not equal 0 on the real line. Firstly, we establish sufficient conditions on the initial data to guarantee that the corresponding solution blows up in finite time for the modified two-component Camassa-Holm (MCH2) system. Then an infinite propagation speed for MCH2 is proved in the following sense: the corresponding solution u(x, t) + kappa with compactly supported initial data (u(0)(x) + kappa, rho(0)(x)) does not have compact x-support in its lifespan.