On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion

Abstract

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schr\"odinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $\Delta^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (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 Schr\"odinger 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 $i\partial _{t}u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$.