## 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