Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Abstract

The current paper devotes to the continuation of the solutions for the Dullin–Gottwald–Holm equation with forcing. Due to forcing, this equation loses the conservation of energy and cannot be regarded as the Hamiltonian system. Thus, through employing the characteristic method and exploiting the balance law, we prove the existence of a semigroup of a global dissipative solution, which is defined for each initial value $u_0\in H^1\left(ℝ\right)$ and depends continuity on it.