Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2024-04-01 , DOI: 10.1007/s00028-024-00961-y Maykel Belluzi
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.
中文翻译：
含时线性算子的抛物型方程的扰动：线性过程和解的收敛
在这项工作中，我们考虑以下形式的抛物线方程
$$\begin{对齐} (u_{\varepsilon })_t +A_{\varepsilon }(t)u_{{\varepsilon }} = F_{\varepsilon } (t,u_{{\varepsilon } }), \结束{对齐}$$其中\(\varepsilon \)是\([0,\varepsilon _0)\)中的参数，而\(\{A_{\varepsilon }(t), \ t\in {\mathbb {R}}\} \)是一个统一的部门运营商家族。作为\(\varepsilon \rightarrow 0^{+}\)，我们假设方程收敛到
$$\begin{对齐} u_t +A_{0}(t)u_{} = F_{0} (t,u_{})。 \end{对齐}$$线性算子\(A_{\varepsilon }(t)\)上发现的时间依赖性意味着线性过程是通过常量公式变分获得解的中心对象。在族\(A_{\varepsilon }(t)\)的适当条件下，当\(\varepsilon \rightarrow 0^{+}\)收敛到\(A_0(t)\)时，我们得到一个 Trotter - 与\(A_{\varepsilon }(t) \) 相关的线性过程\(U_{\varepsilon }(t,\tau ) \) 的加藤型近似定理，估计其收敛于线性过程\(U_0( t,\tau )\)与\(A_0(t)\)关联。通过常量公式的变分，并假设\(F_{\varepsilon }\)收敛于\(F_0\)，我们分析了这种线性过程的收敛性如何转化为半线性方程的解。我们用两个例子来说明这个想法。首先是有界平滑域中的反应扩散方程\(\Omega \subset {\mathbb {R}}^{3}\)
$$\begin{对齐}\begin{对齐}&(u_{\varepsilon })_t - div (a_{\varepsilon } (t,x) \nabla u_{\varepsilon }) +u_{\varepsilon } = f_ {\varepsilon } (t,u_{\varepsilon }), \quad x\in \Omega , t> \tau , \\ \end{aligned} \end{aligned}$$其中\(a_\varepsilon \)收敛到函数\(a_0\)，\(f_{\varepsilon }\)收敛到\(f_0\)。我们在这个例子中应用抽象理论，获得线性过程和解的收敛性。因此，我们还获得了与每个问题相关的回调吸引子族的上半连续性。第二个例子是非自治强阻尼波动方程
$$\begin{对齐} u_{tt}+(-a(t) \Delta _D) u + 2 (-a(t)\Delta _D)^{\frac{1}{2}} u_t = f( t,u), \quad x\in \Omega , t>\tau ,\end{对齐}$$其中\(\Delta _D\)是域\(\Omega \)中具有狄利克雷边界条件的拉普拉斯算子，我们在扰动相关线性算子的分数幂时分析解的收敛性。
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