Blow-up of solutions for the dissipative Dullin–Gottwald–Holm equation with arbitrary coefficients

We study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation describing unidirectional propagation of surface waves in a shallow water regime:ut−α2uxxt+c0ux+3uux+γuxxx+λ(u−α2uxx)=α2(2uxuxx+uuxxx).ut−α2uxxt+c0ux+3uux+γuxxx+λ(u−α2uxx)=α2(2uxuxx+uuxxx). In the present paper we demonstrate the simple conditions on the initial data that lead to blow-up of the solution in finite time.