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

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.