Scattering of an inhomogeneous coupled Schrödinger system in the conformal space

Abstract

Abstract This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ ( ∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ > 0 , j ∈ [ 1 , m ] . i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x ⁢ f ∈ L 2 ⁢ ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0}<p\leq\frac{2-\rho}{N}+1} , where p 0 {p_{0}} is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ ( t ) ∥ L r ⁢ ( ℝ N ) ≲ 1 \lvert t\rvert^{\alpha}\lVert u(t)\rVert_{L^{r}(\mathbb{R}^{N})}\lesssim 1 for some α > 0 {\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}<p<\frac{2-\rho}{N-2}+1} . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 {p\geq 2} , which restricts the space dimensions to N ≤ 3 {N\leq 3} . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 {\rho<0} .