Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 55, pp. 1-21.

Title: Non global solutions for non-radial inhomogeneous nonlinear Schrodinger equations

Authors: Ruobing Bai (Henan Univ., Kaifeng, China)
         Tarek Saanouni (Qassim Univ., Buraydah, Saudi Arabia)

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 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. 


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.