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Energy‐critical scattering for focusing inhomogeneous coupled Schrödinger systems.

Authors :
Ghanmi, Radhia
Saanouni, Tarek
Source :
Mathematical Methods in the Applied Sciences. 7/30/2024, Vol. 47 Issue 11, p9109-9136. 28p.
Publication Year :
2024

Abstract

This work investigates the time asymptotics of the inhomogeneous coupled Schrödinger equations i∂tuj+Δuj=|x|−ρ∑1≤k≤majk|uk|p|uj|p−2uj$$ \mathrm{i}{\partial}_t{u}_j+\Delta {u}_j={\left|x\right|}^{-\rho}\left(\sum \limits_{1\le k\le m}{a}_{jk}{\left|{u}_k\right|}^p\right){\left|{u}_j\right|}^{p-2}{u}_j $$, where j∈[1,m],ρ>0$$ j\in \left[1,m\right],\rho >0 $$, and uj:ℝt×ℝx3→ℂ$$ {u}_j:{\mathrm{\mathbb{R}}}_t\times {\mathrm{\mathbb{R}}}_x^3\to \mathrm{\mathbb{C}} $$. Here, one treats the energy‐critical regime u(0,·)∈[H1(ℝN)]m$$ u\left(0,\cdotp \right)\in {\left[{H}^1\left({\mathrm{\mathbb{R}}}^N\right)\right]}^m $$ and 1=sc:=N2−2−ρ2(p−1)$$ 1={s}_c:= \frac{N}{2}-\frac{2-\rho }{2\left(p-1\right)} $$. This is the index of the invariant Sobolev norm under the dilatation ‖λ2−ρ2(p−1)u(λ2t,λ·)‖H˙sc=λμ−N2+2−ρ2(p−1)‖u(λ2t)‖H˙sc$$ {\left\Vert {\lambda}^{\frac{2-\rho }{2\left(p-1\right)}u}\left({\lambda}^2t,\lambda \cdotp \right)\right\Vert}_{{\dot{H}}^{s_c}}={\lambda}^{\mu -\frac{N}{2}+\frac{2-\rho }{2\left(p-1\right)}}{\left\Vert u\left({\lambda}^2t\right)\right\Vert}_{{\dot{H}}^{s_c}} $$. To the authors knowledge, the technique used in order to prove the scattering of an energy global solution to the above problem in the sub‐critical regime s<sc$$ s<{s}_c $$ is no more applicable for s=sc$$ s={s}_c $$. In order to overcome this difficulty, one uses the Kenig–Merle road map. In order to avoid a singularity of the source term, one considers the case p≥2$$ p\ge 2 $$, which restricts the space dimensions to N≤3$$ N\le 3 $$. Moreover, in order to use the Sobolev injection H˙1↪L2NN−2$$ {\dot{H}}^1\hookrightarrow {L}^{\frac{2N}{N-2}} $$, one restricts the space dimensions to N=3$$ N=3 $$. Compared with the previous work for the first author (Inhomogeneous coupled non‐linear Schrödinger systems. J. Math. Phys. 62, 101508 (2021)), the method consisting on dividing the integrals in the unit ball and its complementary seems not sufficient to conclude in the present study because of the energy‐critical exponent. Instead, one uses some Caffarelli–Kohn–Nirenberg weighted type inequalities. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
01704214
Volume :
47
Issue :
11
Database :
Academic Search Index
Journal :
Mathematical Methods in the Applied Sciences
Publication Type :
Academic Journal
Accession number :
177773306
Full Text :
https://doi.org/10.1002/mma.10062