Geodesic
Mixed problems for the~Korteweg--de~Vries equation
Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 903-935

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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. 
@article{SM_1999_190_6_a6,
     author = {A. V. Faminskii},
     title = {Mixed problems for {the~Korteweg--de~Vries} equation},
     journal = {Sbornik. Mathematics},
     pages = {903--935},
     publisher = {mathdoc},
     volume = {190},
     number = {6},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_6_a6/}
}
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A. V. Faminskii. Mixed problems for the~Korteweg--de~Vries equation. Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 903-935. http://geodesic.mathdoc.fr/item/SM_1999_190_6_a6/