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

Grant number: 13/20297-8
Support Opportunities:Scholarships abroad - Research
Effective date (Start): July 01, 2014
Effective date (End): 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
Research place: 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)

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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)
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, . (13/17636-5, 13/20297-8)
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, . (13/15140-2, 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, . (13/20297-8, 15/16038-2)
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, . (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, . (13/15140-2, 13/17636-5, 13/20297-8, 14/02713-7)

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