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The dynamics of evolution equations governed by fractional powers of closed operators

Grant number: 14/03686-3
Support Opportunities:Scholarships in Brazil - Post-Doctoral
Effective date (Start): September 01, 2014
Effective date (End): August 31, 2015
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Analysis
Principal Investigator:Alexandre Nolasco de Carvalho
Grantee:Flank David Morais Bezerra
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The aim of this research project is to study of the asymptotic behavior of solutions of autonomous and not autonomous evolution equations governed by fractional powers of some closed linear operators. We will consider fractional powers of a linear operator $A:D(A)\subset X\to X$ such that $-A$ is the generator of a linear semigroup with exponential decay in the Banach space $X $, to describe qualitative properties of solutions of problems of the type\begin{equation}\label{Problema3}\displaystyle \frac{du}{dt} + Au = f(u),\ t>0,\quad u(0)=u_0\in X,\end{equation}where the nonlinearity $f$ is a map satisfying suitable growth assumptions. The central idea of the method is to consider fractional approximations of \eqref{Problema3} of the type\begin{equation}\label{Problema4}\displaystyle \frac{du}{dt} + A^\alpha u = f(u),\ t>0,\quad u(0)=u_0\in X,\end{equation}and study the convergence with rate (depending on the power $\alpha \in [0,1]$) of the dynamics of \eqref{Problema4} when $\alpha$ tends to 1.One of the interesting features in this method is that the problem \eqref{Problema4} is more regular than the problem \eqref{Problema3}; that is, even if problem \eqref{Problema3} has hyperbolic structure, its fractional approximations \eqref{Problema4} have parabolic structure.

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Scientific publications (6)
(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)
BEZERRA, F. D. M.; NASCIMENTO, M. J. D.; DA SILVA, S. H.. A class of dissipative nonautonomous nonlocal second-order evolution equations. APPLICABLE ANALYSIS, v. 96, n. 13, p. 2180-2191, . (14/03686-3, 14/03109-6)
BEZERRA, FLANK D. M.; CARVALHO, ALEXANDRE N.; DLOTKO, TOMASZ; NASCIMENTO, MARCELO J. D.. Fractional Schrodinger equation; solvability and connection with classical Schrodinger equation. Journal of Mathematical Analysis and Applications, v. 457, n. 1, p. 336-360, . (14/03686-3, 13/10341-0, 03/10042-0)
BEZERRA, F. D. M.; CARVALHO, A. N.; CHOLEWA, J. W.; NASCIMENTO, M. J. D.. Parabolic approximation of damped wave equations via fractional powers: Fast growing nonlinearities and continuity of the dynamics. Journal of Mathematical Analysis and Applications, v. 450, n. 1, p. 377-405, . (14/03686-3, 14/03109-6, 03/10042-0)
BEZERRA, FLANK D. M.; CARBONE, VERA L.; NASCIMENTO, MARCELO J. D.; SCHIABEL, KARINA. REGULARITY AND UPPER SEMICONTINUITY OF PULLBACK ATTRACTORS FOR A CLASS OF NONAUTONOMOUS THERMOELASTIC PLATE SYSTEMS. PACIFIC JOURNAL OF MATHEMATICS, v. 301, n. 2, p. 395-419, . (14/03686-3, 17/06582-2)
BEZERRA, FLANK D. M.; CARVALHO, ALEXANDRE N.; NASCIMENTO, MARCELO J. D.. FRACTIONAL APPROXIMATIONS OF ABSTRACT SEMILINEAR PARABOLIC PROBLEMS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v. 25, n. 11, p. 4221-4255, . (03/10042-0, 14/03686-3, 17/06582-2)
BEZERRA, FLANK D. M.; CARBONE, VERA L.; NASCIMENTO, MARCELO J. D.; SCHIABEL, KARINA. PULLBACK ATTRACTORS FOR A CLASS OF NON-AUTONOMOUS THERMOELASTIC PLATE SYSTEMS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v. 23, n. 9, p. 3553-3571, . (14/03686-3, 14/03109-6)

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