اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBlowup on an arbitrary compact set for a Schodinger equation with nonlinear source term
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We consider the nonlinear Schrodinger equation on ${mathbb R}^N $, $Nge 1$, begin{equation*} partial _t u = i Delta u + lambda | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} end{equation*} with $lambda in {mathbb C}$ and $Re lambda >0$, for $H^1$-subcritical nonlinearities, i.e. $alpha >0$ and $(N-2) alpha < 4$. Given a compact set $K subset {mathbb R}^N $, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K $ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( Re lambda )^{- frac {1} {alpha }} (-alpha t + A(x) )^{ -frac {1} {alpha } - i frac {Im lambda } {alpha Re lambda } }$, where $Age 0$ vanishes exactly on $ K $, which is a solution of the ODE $u= lambda | u |^alpha u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4].
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