Well-posedness results for a generalized Klein-Gordon-Schrodinger system

We consider the Klein-Gordon-Schrodinger system i partial derivative(t)psi + Delta psi = phi(2)psi - phi psi, (square + 1)phi = -2|psi|(2)phi + |psi|(2) with additional cubic terms and Cauchy data psi(0)=psi(0) is an element of H-s(R-n),phi(0)=phi 0 is an element of H-k(R-n),and(partial derivative t phi)(0)=phi 1 is an element of Hk-1(R-n) in space dimensions n = 2 and n = 3. We prove the local existence, uniqueness, and continuous dependence on the data in Bourgain-Klainerman-Machedon spaces for low regularity data, e.g., for s=-1/18 and k=3/8+E in the case n = 2 and s = 0 and for k=1/2+E in the case n = 3. Global well-posedness in energy space is also obtained as a special case. Moreover, we show the "unconditional" uniqueness in the space psi is an element of C-0([0, T], H-s), phi is an element of C-0([0,T],Hs+1/2) boolean AND C-1([0,T],Hs-1/2), if s > 3/22 for n = 2 and s > 1/2 for n = 3.