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

Villamizar-Roa, Elder J. ; Banquet, Carlos

Electronic journal of differential equations, Tome 2016 (2016)

Résumé

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

$ 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 $i\partial_tu+\delta \Delta^2 u+\lambda|u|^\alpha u=0$.

Détail

Classification : 35Q55, 35A01, 35A02, 35C06
Keywords: fourth-order Schrödinger equation, biharmonic equation, local and global solutions