The purpose of this talk is to present a new existence theorem for the nonlinear Schrodinger equation. The theorem is an answer to the open conjecture.
be a bounded or unbounded domain in
we consider the nonlinear Schrodinger equation
The following conjecture remains open.
Conjecture (Pecher and von Wahl , 1979). p0:=(N+2)/(N-2) is the largest possible exponent for the global existence of strong solutions to (NLS).
However, we can prove the global existence for all exponents beyond their conjecture.
Theorem (). Let . Then for any
there exists a
unique global strong solution
where v(t) is a solution to (NLS) with initial value .