Abstract:

In this paper, we consider the Cauchy problem of a weakly dissipative modified two-component Dullin-Gottwald-Holm (mDGH2) system. The local well-posedness and the global existence are analyzed, which is to prove that blow-up phenomena cannot happen under the condition $\left(\|y_{0}\|_{L^{2}}^{2}+\|\rho_{0}\|^{2}_{L^{2}}\right)^{\frac{1}{2}}<\frac{4\lambda} {3}.$ We derive the precise blow-up scenario, and then provide several criteria guaranteeing the blow-up of the solutions to the weakly dissipative mDGH2 system. It is worth noting that the solution to the weakly dissipative system is not affected by the weakly dissipative term.

Key words: Two-component peakon system, Weakly dissipative, Blow-up