Maykel Belluzi

Universidade Federal de São Carlos, Brazil

Title: Perturbation of parabolic equations with time-dependent linear operators

Abstract: In this work we consider parabolic equations of the form (u_\eps)_t+A_\eps(t)u_\eps=F_\eps(t;u_\eps); where \eps is a parameter in [0 ; \eps_0) and A_\eps is a family of uniformly sectorial operators. As \eps goes to zero, we assume that the equation converges to u_t+A_0(t)u=F_0(t;u). The time-dependence found on the linear operators A_\eps(t) implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family A_\eps(t) and on its convergence to A_0(t) when \eps goes to zero, we obtain a Trotter-Kato type Approximation Theorem for the linear process U_\eps(t;\tau) associated to A_\eps(t), estimating its convergence to the linear process U_0(t;\tau) associated to A_0(t). Through the variation of constants formula and assuming that F_\eps converges to F_0, we analyze how this linear process convergence is transferred to the solution of the semilinear equation. We illustrate the ideas in two examples. First a reaction-diusion equation in a bounded smooth domain and then a nonautonomous fractional strongly damped wave equation. 

Keywords: Nonautonomous parabolic problems, time-dependent linear operators, perturbed problems, convergence of linear process, convergence of solutions.

References [1]Belluzi, M.B. (2024). Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions , Journal of Dynamics and Dierential Equations, Vol. 24.