Non-autonomous weakly damped plate model on time-dependent domains
The authors study the dynamics of the solutions to a weakly damped plate equation on a time-varying domain. Given a time-dependent \textit{expanding} domain \(Q_\tau\subset\mathbb R^n\times(\tau,\infty),\ \tau\in\mathbb R\), the authors consider a weakly damped plate equation: \begin{align*} &\partial_t^2u +\Delta^2u+\partial_tu+\lambda u+f(u)=g(x,t), && \text{ in }\ Q_\tau,\\ & u=\frac{\partial u}{\partial \nu}=0, && \text{ on }\ \Sigma_\tau,\\ & u(x,\tau) = u^0_\tau(x),\quad \partial_t u(x,t)\Big|_{t=\tau} = u^1_\tau(x), && x \in \Omega_\tau,\tag{1} \end{align*} where \(\lambda\in\mathbb R\) is a real parameter, the nonlinearity \(f\in C^1(\mathbb R)\), \(g(x,t)\in L^2_{\mathrm{loc}}(Q_\tau)\) is a external forcing function, \(\nu=(\nu_x,\nu_t)\) is the outward normal to the lateral boundary \(\Sigma_\tau\) of \(Q_\tau \), and \(u^0_\tau\), \(u^1_\tau\) are the initial data. The space-time domain \(Q_\tau\), and its lateral boundary \(\Sigma_\tau\), satisfy \[ Q_\tau = \bigcup_{t>\tau} \{\Omega_t \times{t}\} , \quad\text{and }\ \Sigma _\tau = \bigcup_{t>\tau}\{\Gamma_t \times{t}\}, \] respectively, where \(\Omega_t\in\mathbb R^n\) is a bounded domain. It is assumed that the domain \(Q_\tau\) is expanding, in other words, that \(\Omega_s\subset \Omega_t\) whenever \(s \le t\). Assuming that the domain is time-dependent and \textit{expanding}, and thet the nonlinear term has a critical growth, the authors prove the existence of time-dependent attractors.
scientific article; zbMATH DE number 7451823
scientific article