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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

This work studies the focusing inhomogeneous nonlinear equation of Hartree type і∂tu - Кλu +|х|-τ-|u|p-2)(Jα*|.|-τ|u|P)u = 0, u(t.x): R x R³ → C. Here, the linear Schrödinger operator reads Kλ:= - Δ+ λ/|x|² for some λ > - 1/4. The Riesz potential is Jα (x) = Cα|x|-(3-α) for certain 0 <α< 3. The singular decaying term |x|-τ for some τ > 0 gives an inhomogeneous non-linearity. One considers the inter-critical regime, namely, 1 +2 (1-τ)+α /3 0 because there is no dispersive estimate L¹→ L∞ for λ<0. Contrarily to the homogeneous case τ = 0, for λ> 0, there is a ground state which minimizes the associated Gagliardo-Nirenberg-type estimate. The purpose is to investigate the energy scattering of global solutions under the ground state threshold. One uses the method of Dodson-Murphy based on Tao's scattering criteria and Morawetz estimates. The decay of the inhomogeneous term |x|-τ avoids any spherically symmetric assumption. [ABSTRACT FROM AUTHOR]

This work studies the inhomogeneous Schrödinger coupled system with potential iðtuj - Hvuj = ±|x| -τ (∑m k=1 ajk|uk||p ) |uj|p-2 uj, 1 ≤ j ≤ m. The wave function components read uj: R × R3 → C for 1 ≤ j ≤ m. The inhomogeneous singular term exponent is τ > 0. The source term is energy sub-critical: 1 < p < 3 - τ. Moreover, in order to avoid a singularity of the term |uj| p-2, one assumes that p ≥ 2. The linear Schrödinger operator reads Hv = -Δ + V, where V is a potential satisfying some assumptions which imply that the dispersive and Strichartz estimates hold. The purpose is two-fold. First, we develop a local well-posedness theory in the energy space [H¹v]m:= [{ƒεL²(R3), √{ƒεL²(R³)}]m. Second, we present a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground-state threshold in the inter-critical focusing regime. The scattering is obtained by using the new approach of Dodson-Murphy which is based on Tao's scattering criteria and Morawetz estimates. The novelty here is the presence of the potential V. The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term |·| -τ, and the presence of the potential V. This note naturally extends previous works by the authors about the above problem for V = 0. [ABSTRACT FROM AUTHOR]

This work studies the three space dimensional focusing inhomogeneous Schrödinger equation with inverse square potential i ∂ t u − (− Δ + λ | x | 2 ) u + | x | − 2 τ | u | 2 (q − 1) u = 0 , u (t , x) : R × R 3 → C. The purpose is to investigate the energy scattering of global inter-critical solutions below the ground state threshold. The scattering is obtained by using the new approach of Dodson-Murphy, based on Tao's scattering criteria and Morawetz estimates. This work naturally extends the recent paper by J. An et al. (Discrete Contin. Dyn. Syst., Ser. B 28(2): 1046–1067 2023). The threshold is expressed in terms the non-conserved potential energy. As a consequence, it can be given with a classical way with the conserved mass and energy. The inhomogeneous term | x | − 2 τ for τ > 0 guarantees the existence of ground states for λ ≥ 0 , contrarily to the homogeneous case τ = 0 . Moreover, the decay of the inhomogeneous term enables to avoid any radial assumption on the datum. Since there is no dispersive estimate of L 1 → L ∞ for the free Schrödinger equation with inverse square potential for λ < 0 , one restricts this work to the case λ ≥ 0 . [ABSTRACT FROM AUTHOR]

This work studies the coupled nonlinear fourth-order wave system u ¨ i + Δ 2 ⁢ u i + u i = ± (∑ 1 ≤ j ≤ m a i ⁢ j ⁢ | u j | p ) ⁢ | u i | p - 2 ⁢ u i . The main goal is to develop a local theory in the energy space and to investigate some issues of the global theory. Indeed, using a standard contraction argument coupled with Strichartz estimates, one obtains a local solution in the inhomogeneous Sobolev space (H 1) m for the energy sub-critical regime. Then the local solution extends to a global one in the attractive regime; also in the energy critical case there is a global solution with small data. For a repulsive source term, by using the potential well theory with a concavity argument, the local solution may concentrate in finite time or extend to a global one. Finally, in the inter-critical regime, one proves the existence of infinitely many non-global solutions with data near to the stationary solution. Here, one needs to deal with the coupled source term which gives some technical restrictions such as p ≥ 2 in order to avoid a singularity. This assumption in the inter-critical regime gives a restriction on the space dimension. [ABSTRACT FROM AUTHOR]

The initial value problem for some coupled non-linear wave equations is investigated. In the defocusing case, global well-posedness and illposedness results are obtained. In the focusing sign, the existence of global and non global solutions are discussed via the potential-well theory. Finally, strong instability of standing waves are established. [ABSTRACT FROM AUTHOR]

We study the initial value problem for some defocusing coupled nonlinear fourth-order Schrödinger equations. We show global well-posedness and scattering in the energy space. [ABSTRACT FROM AUTHOR]