Fractional powers of matrix linear operators and fractional approximations of semilinear problems

• Grant number: 22/01439-5
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: October 01, 2022
• End date: October 09, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Everaldo de Mello Bonotto
• Grantee: Maykel Boldrin Belluzi
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 20/14075-6 - Dynamical systems and their attractors under perturbations, AP.TEM

Abstract

In this work we will study semilinear evolution problems of the form$$u_t +\Lambda u = F(t,u), t>\tau; u(\tau) = u_0,$$subjected to perturbations in the power of the linear operator, that is, perturbations given by$$(u_{\gamma})_t +\Lambda^{\gamma} u_{\gamma} = F(t,u_{\gamma}), t>\tau; u_{\gamma}(\tau) = u_0,$$with $\gamma$ in some interval containing $1$. We refer to these perturbations as fractional approximations of the original problem and we are interested in addressing the case where the equations are given in the form of a system and the linear operator $\Lambda = [\Lambda_{ij}]_{1\leq i,j \leq n}$ is represented by a matrix operator. The proposed objective is initially to characterize the fractional powers of $\Lambda$ in terms of its entries $\Lambda_{ij}$, in order to develop a systematic theory for the calculus of $\Lambda^{\gamma}$. This systematic theory will then be applied to study perturbations of semilinear problems via their fractional approximations. We will use these perturbations to analyze the case where the nonlinearity $F$ presents critical growth, in order to obtain local well-posedness for the critical problem. We will also use fractional approximations to study cases where $F$ is not dissipative. We will see how fractional powers impact the dissipativeness condition and we will study situations where, with the appropriate power, a non-dissipative problem can be approximated by a dissipative problem. (AU)