On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion

Abstract

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schrödinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i{\partial }_{t}u+\epsilon \Delta u+\delta Au+\lambda |u{|}^{\alpha }u=0,$ $x\in {\mathbb{R}}^n,$ $t\in \mathbb{R},$ where $A$ represents either the operator ${\Delta }^2$ (isotropic dispersion) or ${\sum }_{i=1}^d{\partial }_{x_i{x}_i{x}_i{x}_i},1\le d<n$ (anisotropic dispersion), and $\alpha ,\epsilon ,\lambda$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(\epsilon =0)$ for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-$L^p$ spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schrödinger equation $i{\partial }_{t}u+\epsilon \Delta u+\delta {\Delta }^{2}u+\lambda |u{|}^{\alpha }u=0$, as $\epsilon$ goes to zero, in $H^2$-norm, to the solutions of the corresponding biharmonic equation $i{\partial }_{t}u+\delta {\Delta }^{2}u+\lambda |u{|}^{\alpha }u=0$.

References in corpus (2)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\ge 9$

Cited by in corpus (1)

• Existence theory of the nonlinear plate equations