Decay estimates for hyperbolic partial differential equations in the L^p-L^q framework

• Grant number: 13/20297-8
• Support Opportunities: Scholarships abroad - Research
• Start date: July 01, 2014
• End date: June 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marcelo Rempel Ebert
• Grantee: Marcelo Rempel Ebert
• Host Investigator: Michael Reissig
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Institution abroad: Technische Universität Bergakademie Freiberg (TU Bergakademie Freiberg), Germany

Abstract

In this project, we are interested in $L^p-L^q$ decay estimates (not necessarily on the conjugate line) in time for linear hyperbolic equations, probably distinguishing between estimates basing on Fourier multipliers localizing to low-frequencies and high-frequencies of the phase space. We plan to apply these estimates to study semi-linear problems. In particular, we are interested in proving results about global existence of the solution, possibly assuming small initial data. Here we plan to understand in which cases the decay rates of solutions to the semi-linear problems coincide with those ones for the corresponding linear problem, and in which other cases a loss of decay appears. Then the question for the exact loss of decay appears. So methods to show optimality should be developed. We plan to study both models with constant coefficients and with time-dependent coefficients as well. In the case of time-dependent coefficients, we will assume suitable regularity and a sufficient control of the oscillations as t goes to infinity. Also, the interaction of the time-dependent coefficients will be studied to avoid bad influence on the asymptotic profile, or to obtain better decay estimates. In a first moment, we will mainly consider wave-type equations, possibly with damping terms, and with nonlocal terms, like fractional powers of the Laplacian. In this way we cover external and structural damping up to the visco-elastic case. Finally, we plan to study higher order equations and, if possible, first-order systems, $p$-evolution equations and problems in an abstract setting. (AU)

Scientific publications

(5)

(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)

D'ABBICCO, M.; EBERT, M. R.. A classification of structural dissipations for evolution operators. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, v. 39, n. 10, p. 2558-2582, JUL 2016. (13/15140-2, 13/20297-8, 14/02713-7)

EBERT, M. R.; KAPP, R. A.; PICON, T.. L-1-L-p estimates for radial solutions of the wave equation and application. Annali di Matematica Pura ed Applicata, v. 195, n. 4, p. 1081-1091, AUG 2016. (13/17636-5, 13/20297-8)

EBERT, MARCELO REMPEL; REISSIG, MICHAEL. Theory of damped wave models with integrable and decaying in time speed of propagation. Journal of Hyperbolic Differential Equations, v. 13, n. 2, p. 417-439, JUN 2016. (13/20297-8)

D'ABBICCO, M.; EBERT, M. R.; PICON, T.. Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, v. 7, n. 2, p. 261-293, JUN 2016. (13/15140-2, 13/17636-5, 13/20297-8, 14/02713-7)

EBERT, M. R.; REISSIG, M.. Regularity theory and global existence of small data solutions to semi-linear de Sitter models with power non-linearity. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 40, p. 14-54, APR 2018. (13/20297-8, 15/16038-2)