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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 C2-boundary . In we consider the nonlinear Schrodinger equation (NLS)

where , the exponent is a constant and u is a complex-valued unknown function. It is known that (NLS) admits a unique global strong solution under the following condition:
• if N=1,2,
• if (Pecher and von Wahl , 1979),
• if (Shigeta , 1986).

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 u(t):=u(x,t) to (NLS) in such that where
v(t) is a solution to (NLS) with initial value .   Next: $B;29MJ88%(B Tohru Okuzono $BJ?@.(B12$BG/(B6$B7n(B5\$BF|(B