School of Mathematics and Statistics
Henan University
Kaifeng 475004, China
email: baimaths@hotmail.com
Department of Mathematics
College of Science
Qassim University
Buraydah, Saudi Arabia
email: t.saanouni@qu.edu.sa
Ruobing Bai, Tarek Saanouni
Abstract:
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 Schrodinger 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.
Submitted March 7, 2025. Published May 26, 2025.
Math Subject Classifications: 35Q55.
Key Words: Inhomogeneous Schrodinger problem;
nonlinear equations; bi-harmonic; inverse square potential;
finite time blow-up.
DOI: 10.58997/ejde.2025.55
Show me the PDF file (453 KB), TEX file for this article.
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Ruobing Bai School of Mathematics and Statistics Henan University Kaifeng 475004, China email: baimaths@hotmail.com |
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Tarek Saanouni Department of Mathematics College of Science Qassim University Buraydah, Saudi Arabia email: t.saanouni@qu.edu.sa |
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