Homogenization of monotone parabolic problems with several temporal scales.
The author studies the asymptotic behaviour, as \(\varepsilon \) goes to \(0\), of the sequence of weak solutions \(u_\varepsilon \) in \(H^1(0,T;H^1_0(\Omega ),H^{-1}(\Omega ))\) of monotone parabolic problems of the type \[ \frac {\partial }{\partial t} u_\varepsilon (x,t) - \nabla \cdot a\left ( x,t,\frac {x}{\varepsilon },\frac {t}{\varepsilon ^{\prime }_1}, \ldots , \frac {t}{\varepsilon ^{\prime }_m};\nabla u_\varepsilon (x,t)\right )=f(x,t)\; \; \text{ in } \Omega \times (0,T) \] \(u_{\varepsilon }(x,0) = u_0(x)\) in \(\Omega \), \(u_{\varepsilon }(x,t)=0\) on \(\partial \Omega \times (0,T)\), where \(\Omega \subset \mathbb {R}^N\) is an open bounded set with the Lipschitz boundary, \(T>0\), and there exist two well-separated spatial scales \(\{1, \varepsilon \}\), the macroscopic and the microscopic, and \((m+1)\) well-separated temporal scale functions \(\{1,\varepsilon ^\prime _1,\ldots , \varepsilon ^{\prime }_m\}\). Assuming that the function \(a\:\Omega \times (0,T)\times \mathbb {R}^N\times \mathbb {R}^m\times \mathbb {R}^N\to \mathbb {R}^N\) satisfies additional continuity, periodicity and growth conditions, it is proved that the sequence \(\{u_{\varepsilon }\}\) weak converges to \(u\) in \(L^2(0,T;H^1_0(\Omega ))\), and the sequence \(\{\nabla u_\varepsilon \)\} \((2,m+1)\)-scale converge to \(\nabla u+\nabla _y u_1\), where \(u\) is the unique solution of the homogenized problem \[ \frac {\partial }{\partial t} u_ (x,t) - \nabla \cdot b\left ( x,t, ;\nabla u (x,t)\right ) = f(x,t)\; \; \text{ in } \Omega \times (0,T), \] \(u (x,0)=u_0(x)\) in \(\Omega \), \(u (x,t)=0\) on \(\partial \Omega \times (0,T)\) and the characterization of \(u_1\) is given in the following cases: the temporal oscillations are slower than the spatial one, the slow spatial and temporal oscillations are in resonance, the temporal oscillations are faster than the spatial one, the rapid spatial and temporal oscillations are in resonance.
scientific article; zbMATH DE number 6121189