Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions

Abstract

In this work we consider parabolic equations of the form \[ (u_{\varepsilon})_t +A_{\varepsilon}(t)u_{{\varepsilon}} = F_{\varepsilon} (t,u_{{\varepsilon} }), \] where $\varepsilon$ is a parameter in $[0,\varepsilon_0)$ and $\{A_{\varepsilon}(t), \ t\in \mathbb{R}\}$ is a family of uniformly sectorial operators. As $\varepsilon \rightarrow 0^{+}$, 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 $A_{\varepsilon}(t)$ implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family $A_{\varepsilon}(t)$ and on its convergence to $A_0(t)$ when $\varepsilon \rightarrow 0^{+}$, we obtain a Trotter-Kato type Approximation Theorem for the linear process $U_{\varepsilon}(t,\tau)$ associated to $A_{\varepsilon}(t)$, 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 $F_{\varepsilon}$ 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-diffusion equation in a bounded smooth domain, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated to each problem. The second example is a nonautonomous strongly damped wave equation and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.