Homogenization of one-dimensional hyperbolic equations with corrector

Abstract

We consider an elliptic differential operator $A_\varepsilon=- \frac{d}{dx} g(\frac x\varepsilon) \frac{d}{dx}$, $\varepsilon > 0$, with a periodic coefficient that acts in $L_2(\mathbb{R})$. We study the behaviour of solutions of the Cauchy problem for the hyperbolic equation $partial_\tau^2 w_\varepsilon (x,\tau)=- (A_\varepsilon w_\varepsilon) (x,\tau)$ as $\varepsilon \to 0$: we find an approximation of the solution $w_\varepsilon ( \cdot ,\tau)$ in the $L_2 (\mathbb{R})$-norm with error $O(\varepsilon^2 )$ with corrector taken into account. Bibliography: 43 titles.