Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Well-Posedness and Smoothing Effect for Nonlinear Dispersive Equations

where α is a real constant with 2α/3 ̸∈ Z and T > 0. In (1), all the parameters are normalized except for α. Equation (1) appears as a mathematical model for nonlinear pulse propagation phenomena in various fields of physics, especially in nonlinear optics (see [54], [27] and [1]). So far, equation (1) without the third order derivative, that is, the cubic NLS equation has attracted much mathema...

Global Well-Posedness for Schrödinger Equations with Derivative

We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in H s , for s > 2/3 for small L 2 data. The result follows from an application of the " I-method ". This method allows to define a modification of the energy norm H 1 that is " almost conserved " and can be used to perform an iteration argument. We also remark that the same argument can be us...

Well posedness and smoothing effect of Schrödinger-Poisson equation

Well-posedness Results for Triply Nonlinear Degenerate Parabolic Equations

We study the well-posedness of triply nonlinear degenerate ellipticparabolic-hyperbolic problem b(u)t − div ã(u, ∇φ(u)) + ψ(u) = f, u|t=0 = u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b, φ and ψ are supposed to be continuous non-decreasing, and the nonlinearity ã falls within the Leray-Lions framework. Some restrictions are imposed on the dependence...

Well-posedness for One-dimensional Derivative Nonlinear Schrödinger Equations

where u = u(t, x) : R → C is a complex-valued wave function, both λ 6= 0 and k > 5 are real numbers. A great deal of interesting research has been devoted to the mathematical analysis for the derivative nonlinear Schrödinger equations [3, 4, 6, 7, 8, 9, 10, 11, 13, 18, 21]. In [13], C. E. Kenig, G. Ponce and L. Vega studied the local existence theory for the Cauchy problem of the derivative non...