2014/02/10 by Carlos Banquet, Banquet, Carlos, Élder J. Villamizar‐Roa +1
Mathematics · Physics and Astronomy · #35A01 #35A02 #35C06 #35Q55 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1402.2193
openalex publication_date 2014/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with the Cauchy problem associated to the nonlinear\nfourth-order Schr "odinger equation with isotropic and anisotropic mixed\ndispersion. This model is given by the equation i\∂ tu+\ε\n\Δ u+\δ A u+\λ|u|^\α u=0, x\∈ \ℝn, t\∈\n\ℝ, where A represents either the operator \Δ2 (isotropic\ndispersion) or \∑i=1d\∂xixixixi, 1\≤ d<n (anisotropic\ndispersion), and \α, \ε, \λ are given real parameters. We\nobtain local and global well-posedness results in spaces of initial data with\nlow regularity, such as weak-Lp spaces. Our analysis also includes the\nbiharmonic and anisotropic biharmonic equation (\ε=0) for which, the\nexistence of self-similar solutions is obtained as consequence of his scaling\ninvariance. In a second part, we investigate the vanishing second order\ndispersion limit in the framework of weak-Lp spaces. We also analyze the\nconvergence of the solutions for the nonlinear fourth-order Schr "odinger\nequation i\∂ tu+\ε \Δ u+\δ \Δ2 u+\λ|u|^\αν=0, as \ε goes to zero, in H2-norm, to the solutions of the\ncorresponding biharmonic equation i\∂ tu+\δ \Δ2ν+\λ|u|^\α u=0.\n