Abstract:

Initial value problem of the mCH-CH equation with cubic and quadratic nonlinearities is studied, which is an important integrable equations obtained by applying the bi-Hamiltonian duality approach to reduce the Gardner equation. Firstly, by constructing the weight function and using the energy method and Gronwall's inequality, the persistence property of the strong solution at infinity was obtained when the initial data decays exponentially or algebraically. Secondly, we prove that the strong solution $ m(x, t) $ has compact support when the initial data $ m_{0}, u_{0} $ has compact support, and the nontrivial solution $ u(x, t) $ no longer has compact support, but has exponential decay property at infinity.

Key words: mCH-CH equation, Persistence property, Propagation speed