Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

Abstract

This paper concerns the homogenization of nonlinear dissipative hyperbolicproblems\begin{gather*}\partial _{tt}u^{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left(\frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}}\right) \nablau^{\varepsilon }\left( x,t\right) \right) \\+g\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}},u^{\varepsilon }\left( x,t\right) ,\nabla u^{\varepsilon }\left( x,t\right)\right) =f(x,t)\end{gather*}where both the elliptic coefficient $a$ and the dissipative term $g$ areperiodic in the $n+m$ first arguments where $n$ and $m$ may attain anynon-negative integer value. The homogenization procedure is performed withinthe framework of evolution multiscale convergence which is a generalizationof two-scale convergence to include several spatial and temporal scales. Inorder to derive the local problems, one for each spatial scale, the crucialconcept of very weak evolution multiscale convergence is utilized since itallows less benign sequences to attain a limit. It turns out that the localproblems do not involve the dissipative term $g$ even though the homogenizedproblem does and, due to the nonlinearity property, an important part of thework is to determine the effective dissipative term. A brief illustration ofhow to use the main homogenization result is provided by applying it to anexample problem exhibiting six spatial and eight temporal scales in such away that $a$ and $g$ have disparate oscillation patterns.