Decay estimates for semilinear hyperbolic equations

• Grant number: 13/15140-2
• Support Opportunities: Research Grants - Young Investigators Grants
• Start date: February 01, 2014
• End date: January 31, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marcello Dabbicco
• Grantee: Marcello Dabbicco
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Associated researchers: Marcelo Rempel Ebert
• Associated scholarship(s): 14/02713-7 - Decay estimates for semilinear hyperbolic equations, BP.JP

Abstract

We plan to study decay estimates for linear hyperbolic equations. We are interested in different types of decay estimates in time, in Sobolev spaces. On the one hand, we will study the setting. On the other hand, we will study estimates, not necessarily on the conjugate line, distinguishing between low-frequencies and high-frequencies estimates. We plan to apply some of the derived estimates to study semilinear problems. In particular, we are interested in proving results of global existence of the solution, possibly assuming small initial data. In the case of small data solution, we will study in which cases the decay rates of the semilinear problems are the same of the linear one, and in which other cases a loss of decay appears. We plan to study both models with constant coefficients and with time-dependent coefficients. In the case of time-dependent coefficients, we will assume suitable regularity and a sufficient control on the oscillations, to guarantee the desired result. 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. We also plan to study higher order equations and possibly first-order systems, evolution equations and problems in abstract setting. (AU)

Scientific publications

(13)

(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, MARCELLO; CHARAO, RUY COIMBRA; DA LUZ, CLEVERSON ROBERTO. SHARP TIME DECAY RATES ON A HYPERBOLIC PLATE MODEL UNDER EFFECTS OF AN INTERMEDIATE DAMPING WITH A TIME-DEPENDENT COEFFICIENT. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 36, n. 5, p. 2419-2447, MAY 2016. (13/15140-2, 14/02713-7)

D'ABBICCO, MARCELLO; LUCENTE, SANDRA; REISSIG, MICHAEL. A shift in the Strauss exponent for semilinear wave equations with a not effective damping. Journal of Differential Equations, v. 259, n. 10, p. 5040-5073, NOV 15 2015. (13/15140-2, 14/02713-7)

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)

BERARDI, MARCO; D'ABBICCO, MARCELLO. A Critical Case for the Spiral Stability for 2 x 2 Discontinuous Systems and an Application to Recursive Neural Networks. Mediterranean Journal of Mathematics, v. 13, n. 6, p. 4829-4844, DEC 2016. (13/15140-2, 14/02713-7)

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, JAN 2017. (13/15140-2, 14/02713-7, 15/16038-2)

D'ABBICCO, MARCELLO. The threshold of effective damping for semilinear wave equations. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, v. 38, n. 6, p. 1032-1045, APR 2015. (13/15140-2)

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., 2017-01-01. (13/17636-5, 15/16038-2, 14/02713-7, 13/15140-2)

D'ABBICCO, MARCELLO. A NOTE ON A WEAKLY COUPLED SYSTEM OF STRUCTURALLY DAMPED WAVES. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, n. SI, p. 320-329, 2015. (13/15140-2, 14/02713-7)

D'ABBICCO, MARCELLO; LUCENTE, SANDRA. NLWE WITH A SPECIAL SCALE INVARIANT DAMPING IN ODD SPACE DIMENSION. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, n. SI, p. 312-319, 2015. (13/15140-2, 14/02713-7)

D'ABBICCO, MARCELLO; JANNELLI, ENRICO. A damping term for higher-order hyperbolic equations. Annali di Matematica Pura ed Applicata, v. 195, n. 2, p. 557-570, APR 2016. (13/15140-2, 14/02713-7)

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)

D'ABBICCO, MARCELLO; LUCENTE, SANDRA. The beam equation with nonlinear memory. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, v. 67, n. 3, JUN 2016. (13/15140-2, 14/02713-7)

D'ABBICCO, MARCELLO; JANNELLI, ENRICO. Dissipative higher order hyperbolic equations. Communications in Partial Differential Equations, v. 42, n. 11, p. 1682-1706, 2017. (13/15140-2, 14/02713-7)