Modified KdV equation with a source term in a bounded domain

A mixed problem for the modified KdV equation in Q = (0,1) × (0,(infinity)) ut+b(t)uux+a(t)uxxx+d(t)u=f(x,t) where a(t)>0, is considered. No restriction on the sign of d(t) is imposed. We prove the existence and uniqueness of strong and weak global solutions as well as the exponential decay of solutions as t(right arrow)+(infinity).