Mixed problems for the Korteweg–de Vries equation

Results are established concerning the non-local solubility and wellposedness 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 $(0,T)\times(-\infty,0)$. 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 $x\to+\infty$ or $x\to-\infty$. Application of this boundary potential enables us in particular to prove the existence of generalized solutions with non-regular boundary values.