Local and global well-posedness for a quadratic Schrodinger system on Zoll manifolds

Author(s)

Nogueira, Marcelo ^[1] ; Panthee, Mahendra ^[1]

Affiliation

^[1] Univ Estadual Campinas, Dept Math, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 494, n. 1 FEB 1 2021.
• Web of Science Citations: 0

Abstract

We consider the initial value problem (IVP) associated to a quadratic Schriidinger system [i partial derivative(t)upsilon +/- Delta(g)upsilon - upsilon = epsilon(1)u (upsilon) over bar, t is an element of R, x is an element of M, i sigma partial derivative(t)u +/- Delta(g)u - alpha u - epsilon(2)/2 upsilon(2), sigma < 0, alpha is an element of R, epsilon(i) is an element of C (i = 1, 2), (upsilon(0), u(0)) = (upsilon(0), u(0)), posed on a d-dimensional compact Zoll manifold M. Considering sigma = theta/beta with theta, beta is an element of [n(2) : n is an element of Z] we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data (upsilon(0), u(0)) is an element of H-s (M) x H-s (M) whenever s > 1/4 when d = 2 and s > d-2/2 when d >= 3. Moreover, in dimensions 2 and 3, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever s >= 1. (C) 2020 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 16/25864-6 - Nonlinear Evolution Equations of Dispersive Type
• Grantee: Mahendra Prasad Panthee
• Support Opportunities: Regular Research Grants