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.
Let
be a bounded or unbounded domain in
with
compact C^{2}-boundary
.
In
we consider the nonlinear Schrodinger equation
(NLS) |
The following conjecture remains open.
Conjecture (Pecher and von Wahl [2], 1979). p_{0}:=(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 ([1]). Let . Then for any
there exists a
unique global strong solution
u(t):=u(x,t) to
(NLS) in
such that
where v(t) is a solution to
(NLS) with initial value
.