| Περίληψη: |
In this paper, we study the Cauchy problem for the nonlinear Hartree equation associated with Dunkl Laplace operator: \[ \begin{cases} i\partial_tu(t,x)+\Delta_ku(t,x)=(|x|^{-\beta}\ast_D|u(t,x)|^2)\cdot u(t,x),\\ u(0,x)=\varphi(x)\in L_k^2(\mathbb{R}^d), \end{cases} \] { i ∂ t u (t , x) + Δ k u (t , x) = (| x | − β ∗ D | u (t , x) | 2) ⋅ u (t , x) , u (0 , x) = φ (x) ∈ L k 2 (R d) , where $ \Delta _k $ Δ k is the Dunkl Laplace operator, and $ \ast _D $ ∗ D denotes the Dunkl convolution. For $ 0 0 < β < min { 2 , d + 2 γ } , we establish both local and global well-posedness in $ L_k^2(\mathbb {R}^d) $ L k 2 (R d) in the mass-subcritical regime by combining Strichartz estimates with convolution estimates in the Dunkl setting. In the mass-critical case $ \beta =2 $ β = 2 , we prove the local theory, and obtain a global solution when the initial value is sufficiently small in $ L_k^2(\mathbb {R}^d) $ L k 2 (R d). Moreover, we establish the scattering behavior in the critical case, and show that if blow-up occurs in finite time, then the $ H_k^s $ H k s -norm necessarily diverges for all s>0. Finally, we develop a long-time perturbation theory which ensures that approximate solutions remain close to exact solutions under control of the critical norm. [ABSTRACT FROM AUTHOR] |