On the nonlocal well-posedness of a mixed problem in a half-strip for the generalized Korteweg-de Vries equation

Following his earlier work on generalized KdV equations [Trudy Moskov. Mat. Obshch. {bf 51} (1988), 54--94, 258; [msn] MR0983632 (90c:35172) [/msn]; Trudy Sem. Petrovsk. No.~13 (1988), 56--105, 256--257; [msn] MR0961429 (90c:35173) [/msn]], the author considers problems of well-posedness and solvability for the generalized KdV equation (*) u_t+u_{xxx}+au_x+(g(u))_x=f(t,x) on the half-strip (0,T)timesBbb R_+ (T>0,Bbb R_+equiv(0,+infty)) with the initial and boundary conditions u(t=0,x)=u_0(x) for xgeq 0 and u(t,x=0)=u_1(t) for 0leq tleq T. Here, a denotes an arbitrary real constant and the function g satisfies the conditions gin C^3(Bbb R) and g(0)=g'(0)=0. Using the additional condition |g(u)|leq c(|u|^{frac{10}{3}}+|u|) (uinBbb R) suggested by a condition first proposed by J. L. Bona and L. Luo [in {it Applied analysis (Baton Rouge, LA, 1996)}, 59--125, Contemp. Math., 221, Amer. Math. Soc., Providence, RI, 1999; [msn] MR1647197 (99m:35201) [/msn]], and taking u_0in H^1(Bbb R_+), u_1in H^1(0,T), u_0(0)=u_1(0), and fin L_2(0,T;H^1(Bbb R_+)), it is proved that equation (*) has a unique generalized solution u(t,x) on a certain Sobolev space X_T related to certain spaces introduced by C. E. Kenig, G. Ponce and L. Vega [Duke Math. J. {bf 59} (1989), no.~3, 585--610; [msn] MR1046740 (91d:35190) [/msn]].