On the low regularity of the modified Korteweg–de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg–de Vries equation ut+uxxx−uxx+(u3)x=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.