We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on R d . These results extend the previous ones in [22] for σ ≥ 2. This covers the well-known result for the Schrödinger equation σ = 2 given in [4]. In the case σ ∈ (0, 2)\{1}, we give the local well-posedness in sub-critical case for all exponent ν > 1 in contrast of ones in [22]. This also generalizes the ones of [11] when d = 1 and of [17] when d ≥ 2 where the authors considered the cubic fractional Schrödinger equation with σ ∈ (1, 2). We also give the global existence in energy space under some assumptions. We finally prove the local well-posedness in sub-critical and critical cases for the pure power-type nonlinear fractional wave equations.

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We consider the focusing nonlinear Schrödinger equation with inverse square poten-Using the profile decomposition obtained recently by the first author [1], we show that in the L 2 -subcritical case, i.e. 0 < α < 4 d , the sets of ground state standing waves are orbitally stable. In the L 2 -critical case, i.e. α = 4 d , we show that ground state standing waves are strongly unstable by blow-up.

We prove the local well-posedness for the nonlinear fourth-order Schrödinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global well-posedness above mass and energy spaces under some assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the super-critical range.

We study the dynamical properties of blowup solutions to the focusing L2-supercritical nonlinear fractional Schrödinger equation i∂tu − (−Δ)su = −|u|αu on [0,+∞)×Rd, where d≥2,d2d−1≤s&lt;1, 4sd&lt;α&lt;4sd−2s, and the initial data u(0)=u0∈Ḣsc∩Ḣs is radial with the critical Sobolev exponent sc. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in Ḣsc∩Ḣs. As a result, we obtain the Ḣsc-concentration of blowup solutions with bounded Ḣsc-norm and the limiting profile of blowup solutions with critical Ḣsc-norm.

We study the asymptotic dynamics for solutions to a system of nonlinear Schr€ odinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virialtype identities.