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非線形 Schrodinger 型方程式の作用素論的扱い

岡沢     登(東京理科大学・理学部)

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 $\Omega$ be a bounded or unbounded domain in $\mathbb R^N$ with compact C2-boundary $\partial\Omega$. In $L^2(\Omega):=L^2(\Omega; {\Bbb
C})$ we consider the nonlinear Schrodinger equation

\dfrac{\partial u}{\partial t}-i\Delta u+\vert...
u(x,0)=u_{0}(x), \quad x \in \Omega,&
\end{displaymath} (NLS)

where $i = \sqrt{-1}$, the exponent $p \ge 1$ 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:

The following conjecture remains open.

Conjecture (Pecher and von Wahl [2], 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 $p \ge 1$beyond their conjecture.

Theorem ([1]). Let $p \ge 1$. Then for any $u_{0}
\in H^2(\Omega) \cap H^1_0(\Omega) \cap L^{2p}(\Omega)$ there exists a unique global strong solution u(t):=u(x,t) to (NLS) in $L^2(\Omega)$ such that
\begin{gather*}u(\cdot) \in L^{\infty}(0, \infty;H^2(\Omega) \cap L^{2p}(\Omega)...
...L^{p+1}}^{p+1} \le 2^{p-1}c(u_0,v_0) \Vert u_0 - v_0
\Vert _{L^2},
v(t) is a solution to (NLS) with initial value $v_0 \in
H^2(\Omega) \cap H^1_0(\Omega) \cap L^{2p}(\Omega)$.

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Tohru Okuzono 平成12年6月5日