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On the critical exponent for the Schrödinger equation with a nonlinear boundary condition. (English) Zbl 1150.35303

Summary: We study the Schrödinger equation: \(\text iu_t+u_{xx}=0\), \(x\in \mathbb R_+\), \(t>0\) with a nonlinear boundary condition \(-u_x(0,t)=| u(0,t)| ^{p-1}u(0,t)\), \(t>0\). We show that if \(1<p<3\), every solution is global in \(H^1(\mathbb R_+)\), while if \(p\geq 3\), then nonglobal solutions exist.

MSC:

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B33 Critical exponents in context of PDEs
35G15 Boundary value problems for linear higher-order PDEs
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