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

In this work, we consider parabolic equations of the form \(\begin{aligned} (u_{\varepsilon })_t +A_{\varepsilon }(t)u_{{\varepsilon }} = F_{\varepsilon } (t,u_{{\varepsilon } }), \end{aligned}\) $$\begin{array}{c}\hfill {(u_{\epsilon })}_t+A_{\epsilon }(t)u_{\epsilon }=F_{\epsilon }(t,u_{\epsilon }),\end{array}$$ where \(\varepsilon \) $\epsilon$ is a parameter in \([0,\varepsilon _0)\) $[0,{\epsilon }_0)$ , and \(\{A_{\varepsilon }(t), \ t\in {\mathbb {R}}\}\) $\{A_{\epsilon }(t),t\in \mathbb{R}\}$ is a family of uniformly sectorial operators. As \(\varepsilon \rightarrow 0^{+}\) $\epsilon \to 0^{+}$ , we assume that the equation converges to \(\begin{aligned} u_t +A_{0}(t)u_{} = F_{0} (t,u_{}). \end{aligned}\) $$\begin{array}{c}\hfill u_t+A_0(t)u_{}=F_0(t,u_{}).\end{array}$$ The time-dependence found on the linear operators \(A_{\varepsilon }(t)\) $A_{\epsilon }(t)$ implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family \(A_{\varepsilon }(t)\) $A_{\epsilon }(t)$ and on its convergence to \(A_0(t)\) $A_0(t)$ when \(\varepsilon \rightarrow 0^{+}\) $\epsilon \to 0^{+}$ , we obtain a Trotter-Kato type Approximation Theorem for the linear process \(U_{\varepsilon }(t,\tau )\) $U_{\epsilon }(t,\tau )$ associated with \(A_{\varepsilon }(t)\) $A_{\epsilon }(t)$ , estimating its convergence to the linear process \(U_0(t,\tau )\) $U_0(t,\tau )$ associated with \(A_0(t)\) $A_0(t)$ . Through the variation of constants formula and assuming that \(F_{\varepsilon }\) $F_{\epsilon }$ converges to \(F_0\) $F_0$ , 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 \(\Omega \subset {\mathbb {R}}^{3}\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ \(\begin{aligned}\begin{aligned}&(u_{\varepsilon })_t - div (a_{\varepsilon } (t,x) \nabla u_{\varepsilon }) +u_{\varepsilon } = f_{\varepsilon } (t,u_{\varepsilon }), \quad x\in \Omega , t> \tau , \\ \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{cc}& {(u_{\epsilon })}_t-div(a_{\epsilon }(t,x)\mathrm{\nabla }{u}_{\epsilon })+u_{\epsilon }=f_{\epsilon }(t,u_{\epsilon }),x\in \mathrm{\Omega },t>\tau ,\hfill \\ \hfill \end{array}\end{array}$$ where \(a_\varepsilon \) $a_{\epsilon }$ converges to a function \(a_0\) $a_0$ , \(f_{\varepsilon }\) $f_{\epsilon }$ converges to \(f_0\) $f_0$ . 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 \(\begin{aligned} u_{tt}+(-a(t) \Delta _D) u + 2 (-a(t)\Delta _D)^{\frac{1}{2}} u_t = f(t,u), \quad x\in \Omega , t>\tau ,\end{aligned}\) $$\begin{array}{c}\hfill u_{\mathit{tt}}+(-a(t){\mathrm{\Delta }}_D)u+2{(-a(t){\mathrm{\Delta }}_D)}^{\frac{1}{2}}{u}_t=f(t,u),x\in \mathrm{\Omega },t>\tau ,\end{array}$$ where \(\Delta _D\) ${\mathrm{\Delta }}_D$ is the Laplacian operator with Dirichlet boundary conditions in a domain \(\Omega \) $\mathrm{\Omega }$ and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.