Blow up of the critical norm for some radial L2 super critical nonlinear Schrödinger equations

• Autores: Frank Merle, Pierre Raphäel
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 130, Nº 4, 2008, págs. 945-978
• Idioma: inglés
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• Resumen

  • We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}< p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with $0< s_c<1$ and this covers the physically relevant case $N=p=3$. The existence of finite time blow up solutions is known. Let $p_c=\frac{N}{2}(p-1)$ so that $\dot{H}^{s_c}\subset L^{p_c}$. Let $u(t)\in \dot{H}^{s_c}\cap \dot{H}^1$ be a radially symmetric blow up solution which blows up at $0