Electronic Journal of Differential Equations (Jan 2016)
On the Schrodinger equations with isotropic and anisotropic fourth-order dispersion
Abstract
This article concerns the Cauchy problem associated with the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $$ i\partial_tu+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,\quad x\in\mathbb{R}^{n},\; t\in \mathbb{R}, $$ where A is 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 real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, based on weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(\epsilon=0)$; in this case, we obtain the existence of self-similar solutions because of their scaling invariance property. In a second part, we analyze the convergence of solutions for the nonlinear fourth-order Schrodinger equation $$ i\partial_tu+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alpha u=0, \quad x\in\mathbb{R}^{n},\; t\in \mathbb{R}, $$ as $\epsilon$ approaches zero, in the $H^2$-norm, to the solutions of the corresponding biharmonic equation $i\partial_tu+\delta \Delta^2 u+\lambda|u|^\alpha u=0$.
