Mixed problems for the Korteweg-de Vries equation

Results are established concerning the non-local solubility and well posedness in various function spaces of the mixed problem for the Korteweg-de Vries equation u(t) + u(xxx) + au(x) + uu(x) = f(t, x) in the half-strip (0,T) x (-infinity,0). Some a priori estimates of the solutions are obtained using a special solution J(t, x) of the linearized Kdv equation of boundary potential type. Properties of J are studied which differ essentially as x --> +infinity or x --> -infinity. Application of this boundary potential enables us in particular to prove the existence of generalized solutions with non-regular boundary values.