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Finite-time blowup for a Schrödinger equation with nonlinear source term (1805.06415v2)

Published 16 May 2018 in math.AP

Abstract: We consider the nonlinear Schr\"odinger equation [ u_t = i \Delta u + | u |\alpha u \quad \mbox{on ${\mathbb R}N $, $\alpha>0$,} ] for $H1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|\alpha U$. In the simplest case, $U(t,x)=(|x|k-\alpha t){-\frac 1\alpha}$ for $t<0$, $x\in {\mathbb R}N$. For $k$ sufficiently large, $U$ satisfies $|\Delta U|\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.

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