Memory effects and homogenization (Q2276719)

For each \(\epsilon >0\) consider problem \(\frac{\partial u^{\epsilon}(x,t)}{\partial t}+a^{\epsilon}(x)u^{\epsilon}(x,t)=f(x,t),\) \(x\in \Omega\), \(t\in (0,T)\); \(u^{\epsilon}(x,0)=0\), \(x\in \Omega\), and suppose that \(0\leq \alpha \leq a^{\epsilon}(x)\leq \beta\), a.e. and f is bounded and measurable. The author proved in another paper [Partial differential equations and the calculus of variations. Essays in Honor of Ennio de Giorgi, 925-938 (1989; Zbl 0682.35028)] that if \(a^{\epsilon}\) converges weakly to a function \(a^ 0\) then a subsequence of \(u^{\epsilon}\) converges weakly to a \(u^ 0\) which satisfies the integral equation \(\frac{\partial u^ 0(x,t)}{\partial t}+a^ 0(x)u^ 0(x,t)-\int^{t}_{0}k(x,t-s)u^ 0(x,s)ds=f(x,t),\) \(x\in \Omega\), \(t\in (0,T)\) \(u^ 0(x,0)=0\), \(x\in \Omega\), where k has a suitable representation. In this paper two different approaches are given to the time-dependent linear case and one of this is used to investigate the nonlinear case.