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In this paper, we investigate the modified two-component Camassa-Holm equation with  κ ≠ 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 ) + κ with compactly supported initial data  ( u 0 ( x ) + κ , ρ 0 ( x ) ) does not have compact  x -support in its lifespan. MSC: 37L05, 35Q58, 26A12.

journal of inequalities and applications

Wave breaking and infinite propagation speed for a modified two-component Camassa-Holm system with "Equation missing"