Electron. J. Diff. Equ., Vol. 2016 (2016), No. 13, pp. 1-20.

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

Elder J. Villamizar-Roa, Carlos Banquet

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

Submitted August 22, 2015. Published January 7, 2016.
Math Subject Classifications: 35Q55, 35A01, 35A02, 35C06.
Key Words: Fourth-order Schrodinger equation; biharmonic equation; local and global solutions.

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Elder J. Villamizar-Roa
Universidad Industrial de Santander
Escuela de Matemáticas
A.A. 678, Bucaramanga, Colombia
email: jvillami@uis.edu.co
Carlos Banquet
Universidad de Córdoba
Departamento de Matemáticas y Estadística
A.A. 354, Montería, Colombia
email: cbanquet@correo.unicordoba.edu.co

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