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Lp-Lq decay estimates for Evolution Operators


In this project, we are interested in Lp-Lq decay estimates (not necessarily on the conjugate line) in time for linear hyperbolicequations or more in general, p-evolution equations. The results are derived by developing a suitable WKB analysis.We plan to apply these estimates to study semi-linear problems. In particular, we are interested in proving results about global existence (in time) 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. 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.; 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)
D'ABBICCO, M.; EBERT, M. R.; LUCENTE, S.. Self-similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, v. 40, n. 18, p. 6480-6494, . (15/16038-2)
D'ABBICCO, MARCELLO; EBERT, MARCELO REMPEL; PICON, TIAGO HENRIQUE. The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, v. 25, n. 3, p. 696-731, . (13/17636-5, 15/16038-2)
D'ABBICCO, M.; EBERT, M. R.. A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 149, p. 1-40, . (13/15140-2, 14/02713-7, 15/16038-2)
D'ABBICCO, MARCELLO; EBERT, MARCELO REMPEL; PICON, TIAGO; DANG, P; KU, M; QIAN, T; RODINO, LG. Global Existence of Small Data Solutions to the Semilinear Fractional Wave Equation. NEW TRENDS IN ANALYSIS AND INTERDISCIPLINARY APPLICATIONS, v. N/A, p. 7-pg., . (13/17636-5, 15/16038-2, 14/02713-7, 13/15140-2)

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