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.