Asymptotic profile of solutions for some evolution partial differential equations and applications

• Grant number: 17/19497-3
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2018
• End date: March 31, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marcelo Rempel Ebert
• Grantee: Marcelo Rempel Ebert
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil

Abstract

In this project, we are interested in the asymptotic behavior (in time) of solutions for some linear hyperbolic equations or more in general, for evolution equations. The results are derived by developing a suitable WKB analysis.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.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.In a first moment, we will mainly consider wave-type equations, possibly with damping terms, and with nonlocal terms, like fractional powers of theLaplacian. In this way we cover external and structural damping up to the visco-elastic case. Finally, we plan to study evolution equations and, if possible, first-order systems. (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)

EBERT, M. R.; GIRARDI, G.; REISSIG, M.. Critical regularity of nonlinearities in semilinear classical damped wave equations. MATHEMATISCHE ANNALEN, OCT 2019. (18/10231-3, 17/19497-3)

EBERT, MARCELO REMPEL; DO NASCIMENTO, WANDERLEY NUNES. A CLASSIFICATION FOR WAVE MODELS WITH TIME-DEPENDENT POTENTIAL AND SPEED OF PROPAGATION. Advances in Differential Equations, v. 23, n. 11-12, p. 847-888, NOV-DEC 2018. (15/23253-7, 17/19497-3)

EBERT, MARCELO R.; DA LUZ, CLEVERSON R.; PALMA, MAIRA F. G.. The influence of data regularity in the critical exponent for a class of semilinear evolution equations. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, v. 27, n. 5, JUL 27 2020. (17/19497-3)

EBERT, M. R.; GIRARDI, G.; REISSIG, M.. Critical regularity of nonlinearities in semilinear classical damped wave equations. MATHEMATISCHE ANNALEN, v. 378, n. 3-4, p. 16-pg., 2019-10-18. (18/10231-3, 17/19497-3)

D'ABBICCO, MARCELLO; EBERT, MARCELO REMPEL. Asymptotic profiles and critical exponents for a semilinear damped plate equation with time-dependent coefficients. ASYMPTOTIC ANALYSIS, v. 123, n. 1-2, p. 1-40, 2021. (17/19497-3)