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

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  • This paper concerns the homogenization of nonlinear dissipative hyperbolic problems \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) \nabla u^{\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 $g$ are periodic in the $n+m$ first arguments where $n$ and $m$ may attain any non-negative integer value. The homogenization procedure is performed within the framework of evolution multiscale convergence which is a generalization of two-scale convergence to include several spatial and temporal scales. In order to derive the local problems, one for each spatial scale, the crucial concept of very weak evolution multiscale convergence is utilized since it allows less benign sequences to attain a limit. It turns out that the local problems do not involve the dissipative term $g$ even though the homogenized problem does and, due to the nonlinearity property, an important part of the work is to determine the effective dissipative term. A brief illustration of how to use the main homogenization result is provided by applying it to an example problem exhibiting six spatial and eight temporal scales in such a way that $a$ and $g$ have disparate oscillation patterns.
    Mathematics Subject Classification: Primary: 35B27; Secondary: 35L15, 35L70.

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