BLOW UP OF SOLUTIONS TO NONLINEAR NEUMANN BOUNDARY VALUE PROBLEM FOR SCHRÓDINGER EQUATIONS

Our purpose is to consider the life span of solutions to the nonlinear Neumann boundary value problem for one dimensional nonlinear Schr & ouml;dinger equation on a half-line {i partial derivative(t)u+(1)/(2)partial derivative(2)(x)u=0, t>0,x is an element of R+, u(0,x)=u(0)(x), x is an element of R+, -partial derivative(x)u(t,0)=g(t,0), where g(t,0)=e-(pi)/(4)i|u(t,0)|(q). We prove that for any q>2, and r>2(q-1), there exists an initial function u0 is an element of(center dot)H(x)(-(2/r-1/2) )such that the maximal existence time is finite.