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

$$\begin{aligned} (u_{\varepsilon })_t +A_{\varepsilon }(t)u_{{\varepsilon }} = F_{\varepsilon } (t,u_{{\varepsilon } }), \end{aligned}$$

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

$$\begin{aligned} u_t +A_{0}(t)u_{} = F_{0} (t,u_{}). \end{aligned}$$

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 with $A_{\varepsilon }(t)$, estimating its convergence to the linear process $U_0(t,\tau )$ associated with $A_0(t)$. 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 $\Omega \subset {\mathbb {R}}^{3}$

$$\begin{aligned}\begin{aligned}&(u_{\varepsilon })_t - div (a_{\varepsilon } (t,x) \nabla u_{\varepsilon }) +u_{\varepsilon } = f_{\varepsilon } (t,u_{\varepsilon }), \quad x\in \Omega , t> \tau , \\ \end{aligned} \end{aligned}$$

where $a_\varepsilon$ converges to a function $a_0$, $f_{\varepsilon }$ converges to $f_0$. We apply the abstract theory in this example, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated with each problem. The second example is a nonautonomous strongly damped wave equation

$$\begin{aligned} u_{tt}+(-a(t) \Delta _D) u + 2 (-a(t)\Delta _D)^{\frac{1}{2}} u_t = f(t,u), \quad x\in \Omega , t>\tau ,\end{aligned}$$

where $\Delta _D$ is the Laplacian operator with Dirichlet boundary conditions in a domain $\Omega$ and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.