Journal of Advances in Mathematics and Computer Science

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime:
                                        u_t − α²u_xxt + c₀u_x + 3uu_x + γu_xxx + λ(u − α²u_xx) = α²(2u_xu_xx + uu_xxx).
Our main conclusion is that on c0 = − γ/α² and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.