Global weak solutions to the weakly dissipative Degasperis–Procesi equation

The global weak solutions for the Degasperis–Procesi equation with weakly dissipative term are investigated. Provided that u0∈H1(R)u0∈H1(R), y0=(1−∂x2)u0∈M(R), suppy0−⊂(−∞,x0) and suppy0+⊂(x0,∞), the existence and uniqueness of global weak solutions for the equation are shown to be true in the space Wloc1,∞(R+×R)∩Lloc∞(R+;H1(R)).