Non global solutions for non-radial inhomogeneous nonlinear Schrodinger equations
This work concerns the inhomogeneous Schrodinger equation $$ \mathrm{i}\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $\lambda>-(N-2)^2/4$. The linear Schr\"odinger operator is $\mathcal{K}...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2025-05-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/55/abstr.html |
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| Summary: | This work concerns the inhomogeneous Schrodinger equation
$$
\mathrm{i}\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad
u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}.
$$
Here, $s\in\{1,2\}$, $N>2s$ and $\lambda>-(N-2)^2/4$.
The linear Schr\"odinger operator is
$\mathcal{K}_{s,\lambda}:= (-\Delta)^s +(2-s)\frac{\lambda}{|x|^2}$,
and the focusing source term can be local or non-local
$$
F(x,u)\in\{|x|^{-2\tau}|u|^{2(q-1)}u,|x|^{-\tau}|u|^{p-2}
\big(J_\alpha *|\cdot|^{-\tau}|u|^p\big)u\}.
$$
The Riesz potential is $J_\alpha(x)=C_{N,\alpha}|x|^{-(N-\alpha)}$,
for certain $0<\alpha<N$. The singular decaying term $|x|^{-2\tau}$,
for some $\tau>0$ gives an inhomogeneous non-linearity.
One considers the inter-critical regime, namely
$1+\frac{2(s-\tau)}N<q<1+\frac{2(s-\tau)}{N-2s}$ and
$1+\frac{2(s-\tau)+\alpha}{N}<p<1+\frac{2(s-\tau)+\alpha}{N-2s}$.
The purpose is to prove the finite time blow-up of solutions with datum
in the energy space, not necessarily radial or with finite variance.
The assumption on the data is expressed in two different ways.
The first one is in the spirit of the potential well method due to
Payne-Sattinger. The second one is the ground state threshold standard
condition. The proof is based on Morawetz estimates and a non-global
ordinary differential inequality.
This work complements the recent paper by Bai and Li [4]
in many directions. |
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| ISSN: | 1072-6691 |