LIMIT BEHAVIOR OF SATURATED APPROXIMATIONS OF NONLINEAR SCHRODINGER-EQUATION

We consider the solution u(epsilon)(t) of the saturated nonlinear Schrodinger equation i partial derivative u/partial derivative t = -DELTAu - \u\4/(N)u + epsilon\u\q-1u and u(0,.) = phi(.), (1)epsilon where N greater-than-or-equal-to 2, epsilon > 0, 1+41N < q < (N + 2)/(N - 2), u : R x R(N) --> C, phi is a radially symmetric function in H-1(R(N)). We assume that the solution of the limit equation is not globally defined in time. There is a T > 0 such that lim(t-->T) parallel-to u(t)parallel-to H-1 = +infinity, where u(t) is the solution of i partial derivative u/partial derivative t = -DELTAu - \u\(4/N)u and u(0,.) = phi(.). For epsilon > 0 fixed, u(epsilon)(t) is defined for all time. We are interested in the limit behavior as epsilon --> 0 of u(epsilon)(t) for t greater-than-or-equal-to T. In the case where there is no loss of mass in u(epsilon) at infinity in a sense to be made precise, we describe the behavior of u(epsilon) as epsilon goes to zero and we derive an existence result for a solution of (1) after the blow-up time T in a certain sense. Nonlinear Schrodinger equation with supercritical exponents are also considered.