Attractors for fully nonlinear parabolic equations and non-autonomous equations

• Grant number: 18/18703-1
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: February 15, 2019
• End date: February 14, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Alexandre Nolasco de Carvalho
• Grantee: Phillipo Lappicy Lemos Gomes
• Supervisor: Carlos Rocha
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Instituto Superior Técnico (IST), Portugal
• Associated to the scholarship: 17/07882-0 - Einstein constraints and differential equations on the sphere, BP.PD

Abstract

For the research period abroad, I will study dynamical systems in infinite dimensions. I pretend to continue the studies that started in my Phd thesis at FU Berlin in 2017 and started at ICMC-USP in the same year. We will consider two problems described below. For such, I pretend to visit IST-Lisboa in order to contribute with the mathematics being done in Brazil, and increase the research relations with Portugal scientists. The first problem is to study the global attractor for fully nonlinear parabolic equations. The first steps were done in this posdoc project in Brazil: the construction of a Lyapunov function for fully nonlinear equations. Hence, we can decompose the attractor as equilibria points and their connections. We have to compute such connections, proving a Morse-Smale property, and construct a permutation capable of describing the attractor. The second problem pursues the understanding of the global attractor for non-autonomous equations. For such, still little is known about the geometry of the attractors. A central point is to decompose the attractor into smaller invariant sets and its connections, yielding a Morse decomposition. For example, is it possible to show a Poincaré-Bendixson type theorem? (AU)

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)

LAPPICY, PHILLIPO; DANIEL, VICTOR HUGO. Chaos in spatially homogeneous Horava-Lifshitz subcritical cosmologies. Classical and Quantum Gravity, v. 39, n. 13, p. 18-pg., 2022-07-07. (18/18703-1, 20/07532-1, 17/07882-0)

LAPPICY, PHILLIPO. Sturm attractors for fully nonlinear parabolic equations. REVISTA MATEMATICA COMPLUTENSE, v. N/A, p. 23-pg., 2022-08-03. (18/18703-1, 17/07882-0)

DAI, JIA-YUAN; LAPPICY, PHILLIPO. Ginzburg-Landau Patterns in Circular and Spherical Geometries: Vortices, Spirals, and Attractors. SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, v. 20, n. 4, p. 1959-1984, 2021. (18/18703-1, 17/07882-0)