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\(L^p\)-type pullback attractors for a semilinear heat equation on time-varying domains. (English) Zbl 1344.35071

The authors consider the following semilinear parabolic equation with homogeneous Dirichlet boundary condition, that we call (PE): \[ \frac{\partial u}{\partial t}-\Delta u+g(u)=f(t),\quad x \in Q_{\tau} \]
\[ u=0, \quad x \in \Sigma_{\tau};\quad\quad u(\tau,x)=u_{\tau}(x),\quad x \in \mathcal{O}_{\tau}, \] where \(\tau \in \mathbb{R}\), \(u_{\tau}:\mathcal{O}_{\tau}\rightarrow \mathbb{R}\) and \(f:Q_{\tau} \rightarrow \mathbb{R}\) are given. \(g \in C^{1}(\mathbb{R},\mathbb{R})\) is a given function for which it is assumed that there exist non-negative constants \(\alpha_1,\alpha_2\), \(\beta, l\) and \(q \geq 2\) such that \[ -\beta +\alpha_1|s|^{q} \leq g(s)s \leq \beta+\alpha_2|s|^{q},\quad \forall s \in \mathbb{R} \] and \[ g'(s) \geq -l,\quad \forall s \in \mathbb{R}. \] It is assumed that \(f \in L^2_{\mathrm{loc}}(\mathbb{R};H^{-1}(\mathcal{O}_t))\) and that \(\mathcal{O}\) is a non-empty bounded open subset \(\mathbb{R}^{N}\) with a boundary of class \(C^2\) and \(N \leq 2q/(q-2)\).
\(r \in C^{1}(\overline{\mathcal{O}}\times \mathbb{R},\mathbb{R}^N)\) satisfies \(r(\cdot,t):\mathcal{O} \rightarrow \mathcal{O}_t\) is a \(C^2\)-diffeomorphism for all \(t \in \mathbb{R}\). \[ Q_{\tau,T}:=\bigcup_{t \in (\tau,T)}\mathcal{O}_t \times {t} \] for any \(T>\tau\) and \[ Q_{\tau}:=\bigcup_{t \in (\tau,+\infty)}\mathcal{O}_t \times {t},\quad \tau \in \mathbb{R}. \] The inverse of \(r(\cdot,t)\) is supposed to be in the set \(C^{2,1}(\overline{Q}_{\tau,T};\mathbb{R}^N)\), \(\forall \tau<T\).
The main theorem proved in the paper is stated next under the previous assumptions on \(r, \overline{r}\), \(\partial \mathcal{O}\), and also \(r \in C_{b}(\overline{\mathcal{O}} \times \mathbb{R}; \mathbb{R}^N)\), and \[ \int_{-\infty}^{t}e^{\lambda s}\|f(s)\|_{H^{-1}(\mathcal{O}_s)}^{2}\;ds<\infty,\quad \forall t \in \mathbb{R}, \] where \(\lambda\) is the first eigenvalue of \(-\Delta\) in \(H_{0}^{-1}(\Omega)\) with \(\Omega:=\cup_{t \in \mathbb{R}} \mathcal{O}_t\).
The main result states that letting \(U(t,\tau)\) be the process generated by the weak solutions of (PE) and \[ \hat{\mathcal{A}}:=\{\mathcal{A}(t):t \in \mathbb{R}\} \] be the \((L^2,L^2)\) pullback \(\mathcal{D}_{\lambda}\)-attractor of \(U(t,\tau)\). Then, for any \(\delta \geq 0\) and any \(\hat{D}=\{D(t):t \in \mathbb{R}\} \in \mathcal{D}_{\lambda}\), the following properties hold:
(i) \(\hat{\mathcal{A}}\) is \(L^{2+\delta}\)-pullback \(\mathcal{D}_{\lambda}\)-attracting. (ii) there exist two positive constants \(T(t,\delta, \hat{D},\hat{\mathcal{A}})\) (depending only on \(t\), \(\delta\), \(\rho_{\hat{D}}\) and \(\rho_{\hat{\mathcal{A}}}\)) and \(M_{\delta}(t)\) (depending only on \(t\), \(\delta\), \(N\), \(|\mathcal{O}_t|\) and \(\int_{-\infty}^{t}e^{\lambda s}\|f(s)\|_{H^{-1}(\mathcal{O}_s)}^{2}\;ds\)) such that \[ \int_{\mathcal{O}_t}\left|U(t,\tau)u_{\tau}-v(t)\right|^{2+\delta}\;dx \leq M_{\delta}(t)\quad \forall t-\tau \geq T(t,\delta,\hat{D}, \hat{\mathcal{A}}), \] where \(v(\tau) \in \mathcal{A}(\tau)\) \((\tau \in \mathbb{R})\) is an (arbitrary) fixed complete trajectory of \(U(t,\tau)\).

MSC:

35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
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