Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition
The author studies parametrized families of processes \[ {\mathcal U}_\sigma(t,\tau),\quad t\geq \tau,\;\sigma\in\Sigma\tag{1} \] which are associated with nonautonomous semilinear parabolic equations. Here, \({\mathcal U}_\sigma(t,\tau)\) is a bounded operator, acting on a Banach space \(X\) such that: \[ \begin{aligned} & {\mathcal U}_\sigma(t,s){\mathcal U}_\sigma(s,\tau)={\mathcal U}_\sigma(t,\tau),\quad t\geq s\geq \tau,\;\sigma\in\Sigma\tag{a}\\ & U_\sigma(s,s)=I,\quad s\in \mathbb R,\;\sigma\in\Sigma.\tag{b}\end{aligned} \] The symbol space \(\Sigma\) is a metric space with metric \(d(\;)\), on which a translation semigroup \(T(\;)\) acts according to: \[ {\mathcal U}_{T(s)\sigma}(t,\tau)={\mathcal U}_\sigma(t+s,\tau+s),\quad t\geq\tau,\;\sigma\in\Sigma,\;s\geq 0.\tag{2} \] The author recalls a series of definitions related to parametrized processes. Definition 2.3 eg. introduces the notion of an attractor of a parametrized process via (1), uniform with respect to \(\sigma\in\Sigma\). A new concept is introduced by Definition 3.1 in which the notion of closed parametrized family of processes is defined. The notion of closed translation semigroup \(T(s)\), \(s\geq 0\) is then defined by Definition 3.2. The author then proceeds to the proof of some theorems concerning this abstract setting. Theorem 3.5 eg. asserts the existence of an attractor for the parametrized process (1), uniform with respect to \(\sigma\in\Sigma\), provided that \(\Sigma\) is compact, the process (1) and the semigroup \(T(s)\), \(s\geq 0\) closed, and some assumptions of technical nature satisfied. In sect. 5 the abstract theory is applied to a non-autonomous reaction-diffusion equation with dynamic boundary conditions, ie.: \[ \begin{alignedat}{2} & u_t-\Delta u+f(u)=h(t),&\quad & \text{on }[\tau,T]\times \Omega\tag{3}\\ & u_t+\frac{\partial u}{\partial n}+g(u)=0,&\quad & \text{on }(\tau,T)\times \partial\Omega,\\ &u(x,\tau)=u_\tau(x) &\quad & \text{on }\Omega\subseteq R^n.\end{alignedat} \] Here \(\Omega\) is a smooth bounded domain, \(h(t)\in{\mathcal L}^2_{\text{loc}}(R:{\mathcal L}^2(\Omega))\), with \(f,g\in C^1(R,R)\) are subject to some growth conditions. Equation (3) is then cast into a suitable functional frame which allows the application of the abstract theory. Theorem 5.1 now asserts existence and uniqueness of weak solutions of (3). In order to associate a labeled process with (3) on the basis of Theorem 5.1, one takes for \(\Sigma\) the weak closure in \({\mathcal L}^2_{\text{loc}}(R:{\mathcal L}^2(\Omega))\) of the set of translates \(h_0(\cdot,+d)\), \(d\in R\) of a fixed element \(h_0\in{\mathcal L}^2_b(R:{\mathcal L}^2(\Omega))\). Based on this stipulation and on Thm. 5.1 a parametrized process (1) is associated with (3) along established lines. The authors now obtain several theorems in the context. We note Theorem 5.8 which asserts the existence of a compact attractor for (3), uniform with respect to \(\sigma\in\Sigma\).
scientific article; zbMATH DE number 5577466
scientific article