Wave equations with time-dependent speed of propagation, mass and dissipation.

• Grant number: 15/23253-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: March 01, 2016
• End date: December 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Lucas Catão de Freitas Ferreira
• Grantee: Wanderley Nunes Do Nascimento
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

In this project we are interested in estimates of dispersive and Strichartz type for linear wave equations with time-dependent propagation of speed, mass and dissipation. We will assume suitable regularity and a sufficient control of the oscillations as $t\to\infty$. The interaction of the time-dependent coefficients will be analyzed to control its bad influence on the asymptotic profile or to obtain better decay estimates. An approach consists in developing a suitable WKB analysis and using the $L^p-L^q$-framework. Moreover, we will explore other regularity degrees and potentials by means of harmonic analysis spaces such as Marcinkiewicz, Herz, Triebel-Lizorkin and Besov, among others. We plan to use these estimates to study nonlinear problems. In particular, we are interested in proving results about global-in-time existence of solutions (possibly assuming smallness on data) and blow-up and in investigating equations with potentials having multiple singularities. So, methods to show optimality should be developed.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

EBERT, MARCELO REMPEL; DO NASCIMENTO, WANDERLEY NUNES. A CLASSIFICATION FOR WAVE MODELS WITH TIME-DEPENDENT POTENTIAL AND SPEED OF PROPAGATION. Advances in Differential Equations, v. 23, n. 11-12, p. 847-888, NOV-DEC 2018. (15/23253-7, 17/19497-3)