Decay estimates for semilinear hyperbolic equations

• Grant number: 14/02713-7
• Support Opportunities: Scholarships in Brazil - Young Researchers
• 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 research grant: 13/15140-2 - Decay estimates for semilinear hyperbolic equations, AP.JP

Abstract

Neste projeto pretende-se obter estimativas a priori para equa\c{c}\~oes hiperb\'olicas lineares em espa\c{c}os de Sobolev. Espera-se ser necess\'ario utilizar t\'ecnicas diferentes para frequencias pequenas e grandes, obtendo diferentes tipos de decaimento no tempo em cada regi\~ao do espa\c{c}o de fase. Na sequ\^encia, pretende-se realizar aplica\c{c}\~oes das estimativas obtidas para problemas semilineares. Em particular, provar a exist\^encia de solu\c c\~ao global no tempo, uma vez assumido dados iniciais pequenos. Tamb\'em analisar em quais casos o decaimento da solu\c c\~ao do problema semilinear permace o mesmo do problema linear associado. Pretende-se estudar modelos com coeficientes constantes ou vari\'aveis no tempo. No \'ultimo caso, espera-se que condi\c{c}\~oes sobre regularidade e controle nas oscila\c{c}\~oes deva ser impostas para obter-se boas estimativas de decaimento. Num primeiro momento, pretende-se considerar equa\c{c}\~oes de onda, possivelmente com um termo de dissipa\c{c}\~ao, que pode ser na forma de um operador n\~ao local, como uma pot\^encia fracion\'aria do laplaciano. Tamb\'em tem-se interesse em estudar equa\c{c}\~oes de ordem superior e sistemas de equa\c{c}\~oes de primeira ordem, equa\c{c}\~oes de evolu\c{c}\~ao e modelos abstratos.

Scientific publications

(12)

(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. 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; 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; 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, 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)

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)