Pullback attractors for a semilinear heat equation on time-varying domains
The authors study a semilinear heat equation on a domain which varies with time. To be more explicit, let \({\mathcal O}\subseteq \mathbb R^N\) be nonempty bounded, with \(C^2\)-boundary \(\partial{\mathcal O}\), let \(r\in C^1(\overline{\mathcal O}\times \mathbb R : \mathbb R^N)\) be a vector function such that \[ r(\cdot,t):{\mathcal O}\to {\mathcal O}_t\tag{1} \] is a \(C^2\)-diffeomorphism for all \(t\in \mathbb R\). For \(\tau<T\) set \[ \begin{alignedat}{2}2 Q_{\tau T} &=\bigcup_{t\in(\tau,T)}{\mathcal O}_t\times \{t\}, &\quad Q_\tau&= Q_{\tau\infty}\\ \Sigma_{\tau T} &=\bigcup_{t\in(\tau,T)}\partial{\mathcal O}_t\times \{t\}, &\quad \Sigma_\tau&=\Sigma_{\tau\infty}.\end{alignedat} \] We also set \(F(\cdot,t)=r^{-1}(\cdot,t)\). The equation in question is \[ \partial u_t-\Delta u+g(u)=f(t)\text{ in }Q_\tau, \quad u=0\text{ on }\Sigma_\tau, \quad\text{and}\quad u(\tau,x)=u_\tau(x),\qquad x\in{\mathcal O}_\tau.\tag{2} \] \((2)_T\) is the problem obtained from (2) by replacing \(Q_\tau\) and \(\Sigma_\tau\) by \(Q_{\tau T}\) and \(\Sigma_{\tau T}\), respectively. Here \(u_\tau:{\mathcal O}_\tau\to\mathbb R\), \(f:Q_\tau\to \mathbb R\) and \(g\in C^1(R,R)\), such that \[ -\beta+\alpha_1|s|^p\leq sg(s)\leq \beta+\alpha_2|s|^p, \] some \(\beta,\alpha_1,\alpha_2\geq 0\), \(p\geq 2\), and \(g'(s)\geq -\ell\), for \(s\in R\) and some \(\ell\geq 0\). In order to interprete (2), \((2)_T\) in a suitable functional setting, a considerable amount of preparatory steps are needed. Among others, strong and weak solutions are defined. The first major result is provided by Theorem 4.4 which asserts the following. Let \(f\in{\mathcal L}^2_{\text{loc}}(\mathbb R;{\mathcal L}^2({\mathcal O}_t))\). Let either \(\partial{\mathcal O}\) be \(C^2\) and \(N\leq 2p/p-2\) or let else \(\partial{\mathcal O}\) be \(C^m\) with \(m\geq N(p-2)/2p\). Then, for any \(u_\tau\in H^1_0({\mathcal O}_\tau)\cap{\mathcal L}^p({\mathcal O}_\tau)\) and \(\tau<T<\infty\) there exists a unique strong solution of \((2)_T\) which satisfies an energy equally. In the proof, one transforms equation (2) into another equation, say \((2)^*\), defined on the time-independent domain \(\mathcal O\). A similar result (Thm. 5.2) holds for weak solutions of \((2)_T\). It states that if \(f\in{\mathcal L}^2_{\text{loc}}(R:H^{-1}({\mathcal O}_t))\) and \(u_\tau\in{\mathcal L}^2({\mathcal O}_\tau)\) then a unique weak solution \(u(t)\) for equation \((2)_T\) exists for any \(\tau<T\). In addition, \(u(t)\) satisfies some energy inequalities. These results extend to equation (2) (Theorem 6.2). Let \(U(t,\tau)\), \(\tau\leq t\) be the solution operators induced by the weak solutions \(u(\tau,t,\cdot)\) of (2). \(U(t,\tau)\), \(\tau\leq t\) is a process, which extends the notion of solution semigroup from the autonomous to the nonautonomous case. The paper concludes with a proof that the process \(U(t,\tau)\), \(\tau\leq t\) admits a pullback attractor, the nonautonomous counterpart to the familiar attractor in the autonomous case.
scientific article; zbMATH DE number 5563803
scientific article