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Blow up of the critical norm for some radial L^2 super critical nonlinear Schrodinger equations | | Frank Merle
; Pierre Raphael
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15 May 2006 | | Subject: | Analysis of PDEs; Mathematical Physics | | Abstract: | We consider the nonlinear Schr"odinger equation $iu_t=-Delta u- u ^{p-1}u$ in dimension $Ngeq 3$ in the $L^2$ super critical range $frac{N+3}{N-1}leq p<frac{N+2}{N-2}$. The corresponding scaling invariant space is $dot{H}^{s_c}$ with ${1/2}leq s_c<1$ and this covers the physically relevant case N=3, $p=3$. The existence of finite time blow up solutions is known. Let $u(t)in dot{H}^{s_c}cap dot{H}^1$ be a radially symmetric blow up solution which blows up at $0<T<+infty$, we prove that the scaling invariant $dot{H}^{s_c}$ norm also blows up with a lower bound $ u(t) _{dot{H}^{s_c}}geq log(T-t) ^{C_{N,p}} $ as $t o T$. | | Source: | arXiv, math/0605378 | | Services: | Forum | Review | PDF | Favorites | |
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