Singular limit of differential systems with memory. (English) Zbl 1100.35018
Let \(\Omega\) be a smooth bounded domain in \(\mathbb R^3\), \(\omega\in[0,1)\) and \(k\): \([0,\infty)\to \mathbb R\) be a sufficiently smooth nonnegative memory kernel such that \(\int_0^{\infty} k(s)\,ds =1.\) For \(\varepsilon \in (0,1],\) set \(k_{\varepsilon}(s)=\frac{1}{\varepsilon}k(\frac{s}{\varepsilon}).\)
The aim of the paper is to prove, in the framework of dynamical systems, the convergence in an appropriate sense, as \(\varepsilon\to 0,\) of \(u_{\varepsilon}(x,t)\), \(x\in \Omega\), \(t\in \mathbb R\) solution of \(p_{\varepsilon}\) : \[ u_t-\omega\Delta u-(1-\omega)\int_0^{\infty}k_{\varepsilon}(s)\Delta u(t-s)\,ds +\varphi(u)=f,\quad t>0 \] to \(u(x,t)\) solution of \(p_0\): \[ u_t-\Delta u +\varphi(u)=f,\;t>0 \] with Dirichlet boundary conditions on the boundary of \(\Omega.\) Here \(\varphi\) is a suitable nonlinearity and \(f\) is a time independent source term. Let \(A=-\Delta\) on \(L^2(\Omega)\) with domain \(\mathcal D(A)=H^1_0(\Omega)\cap H^2(\Omega),\;H^r=\mathcal D(A^{\frac{r}{2}}),r\in \mathbb R.\)
Following C. Dafermos [Arch. Ration. Mech. Anal. 37, 297–308 (1970; Zbl 0214.24503)] and M. Grasselli and V. Pata [Prog. Nonlinear Differ. Equ. Appl. 50, 155–178 (2002; Zbl 1039.34074)], the authors introduce, under additional assumptions on \(k,\varphi,f,\) \((\varphi(x)=x^3-x\) is allowed), the auxiliary variable \(\eta^t(x,s)=\int_0^s u(x,t-y)\,dy,\) the functions \(\mu (s) =-(1-\omega)k'(s),\mu_{\varepsilon}(s)=\frac{1}{\varepsilon^2}\mu (\frac{s}{\varepsilon}),\) the Hilbert spaces \( \mathcal M^{r}_{\varepsilon}=L^2_{\mu_{\varepsilon}}(\mathbb R^+,H^{r+1})\) and \(\mathcal H^r_{\varepsilon}=H^r\times \mathcal M^r_{\varepsilon}\) for \(\varepsilon >0\), \(\mathcal H^r_0= H^r\). The correct reformulation of \(p_{\varepsilon}\) [resp. \(p_0\)] is \(P_{\varepsilon}\): find \((u_{\varepsilon},\eta_{\varepsilon})\in C([0,\infty),\mathcal H ^0_{\varepsilon})\) solution to \[ u_t+\omega Au+\int_0^{\infty}\mu_{\varepsilon}(s)A\eta(s)\,ds+\varphi(u)=f,\quad \partial_{t}\eta=-\partial_s\eta +u \] for \(t>0\), associated with the initial condition \((u_0,\eta_0)\in \mathcal H^o_{\varepsilon},\) [resp. \(P_0\): find \(u\in C([0,\infty),\mathcal H_0^0)\) solution to \(u_t+Au+\varphi(u)=f,\quad t>0,\;u(0)=u_0.]\) The main result is the existence, for \(\varepsilon \geq 0\), of a strongly continuous semigroup \(S_{\varepsilon}(t)\), on \(\mathcal H^0_{\varepsilon},\) corresponding to \(P_{\varepsilon}\). Sharp estimates, as \(\varepsilon \to 0\), for \(\| u_{\varepsilon}-u\|\) and \(\|\eta_{\varepsilon}\|\), in different functional spaces, are obtained separately when \(\omega \not =0\) (the Coleman-Gurtin case) and \(\omega=0\) (the Gurtin-Pipkin case). Existence, asymptotic behavior of uniform and exponential attractors are also investigated.
The aim of the paper is to prove, in the framework of dynamical systems, the convergence in an appropriate sense, as \(\varepsilon\to 0,\) of \(u_{\varepsilon}(x,t)\), \(x\in \Omega\), \(t\in \mathbb R\) solution of \(p_{\varepsilon}\) : \[ u_t-\omega\Delta u-(1-\omega)\int_0^{\infty}k_{\varepsilon}(s)\Delta u(t-s)\,ds +\varphi(u)=f,\quad t>0 \] to \(u(x,t)\) solution of \(p_0\): \[ u_t-\Delta u +\varphi(u)=f,\;t>0 \] with Dirichlet boundary conditions on the boundary of \(\Omega.\) Here \(\varphi\) is a suitable nonlinearity and \(f\) is a time independent source term. Let \(A=-\Delta\) on \(L^2(\Omega)\) with domain \(\mathcal D(A)=H^1_0(\Omega)\cap H^2(\Omega),\;H^r=\mathcal D(A^{\frac{r}{2}}),r\in \mathbb R.\)
Following C. Dafermos [Arch. Ration. Mech. Anal. 37, 297–308 (1970; Zbl 0214.24503)] and M. Grasselli and V. Pata [Prog. Nonlinear Differ. Equ. Appl. 50, 155–178 (2002; Zbl 1039.34074)], the authors introduce, under additional assumptions on \(k,\varphi,f,\) \((\varphi(x)=x^3-x\) is allowed), the auxiliary variable \(\eta^t(x,s)=\int_0^s u(x,t-y)\,dy,\) the functions \(\mu (s) =-(1-\omega)k'(s),\mu_{\varepsilon}(s)=\frac{1}{\varepsilon^2}\mu (\frac{s}{\varepsilon}),\) the Hilbert spaces \( \mathcal M^{r}_{\varepsilon}=L^2_{\mu_{\varepsilon}}(\mathbb R^+,H^{r+1})\) and \(\mathcal H^r_{\varepsilon}=H^r\times \mathcal M^r_{\varepsilon}\) for \(\varepsilon >0\), \(\mathcal H^r_0= H^r\). The correct reformulation of \(p_{\varepsilon}\) [resp. \(p_0\)] is \(P_{\varepsilon}\): find \((u_{\varepsilon},\eta_{\varepsilon})\in C([0,\infty),\mathcal H ^0_{\varepsilon})\) solution to \[ u_t+\omega Au+\int_0^{\infty}\mu_{\varepsilon}(s)A\eta(s)\,ds+\varphi(u)=f,\quad \partial_{t}\eta=-\partial_s\eta +u \] for \(t>0\), associated with the initial condition \((u_0,\eta_0)\in \mathcal H^o_{\varepsilon},\) [resp. \(P_0\): find \(u\in C([0,\infty),\mathcal H_0^0)\) solution to \(u_t+Au+\varphi(u)=f,\quad t>0,\;u(0)=u_0.]\) The main result is the existence, for \(\varepsilon \geq 0\), of a strongly continuous semigroup \(S_{\varepsilon}(t)\), on \(\mathcal H^0_{\varepsilon},\) corresponding to \(P_{\varepsilon}\). Sharp estimates, as \(\varepsilon \to 0\), for \(\| u_{\varepsilon}-u\|\) and \(\|\eta_{\varepsilon}\|\), in different functional spaces, are obtained separately when \(\omega \not =0\) (the Coleman-Gurtin case) and \(\omega=0\) (the Gurtin-Pipkin case). Existence, asymptotic behavior of uniform and exponential attractors are also investigated.
Reviewer: Denise Huet (Nancy)
MSC:
35B41 | Attractors |
35B25 | Singular perturbations in context of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
35K57 | Reaction-diffusion equations |
37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
45K05 | Integro-partial differential equations |
35K55 | Nonlinear parabolic equations |