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
where A is either the operator
(isotropic dispersion)
or
,
(anisotropic dispersion), and
are real
parameters. We obtain local and global well-posedness results in
spaces of initial data with low regularity, based on weak-
spaces.
Our analysis also includes the biharmonic and anisotropic biharmonic
equation
; 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
as
approaches zero, in the
-norm,
to the solutions of
the corresponding biharmonic equation
.
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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