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.