Homogenization of initial boundary value problems for parabolic systems with periodic coefficients

Abstract

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider matrix elliptic second order differential operators $\mathcal{A}_{D,\varepsilon}$ and $\mathcal{A}_{N,\varepsilon}$ with the Dirichlet or Neumann boundary condition on $\partial \mathcal{O}$, respectively. Here $\varepsilon>0$ is the small parameter. The coefficients of the operators are periodic and depend on $\mathbf{x}/\varepsilon$. The behavior of the operator $e^{-\mathcal{A}_{\dag ,\varepsilon}t}$, $\dag =D,N$, for small $\varepsilon$ is studied. It is shown that, for fixed $t>0$, the operator $e^{-\mathcal{A}_{\dag ,\varepsilon}t}$ converges in the $L_2$-operator norm to $e^{-\mathcal{A}_{\dag}^0 t}$, as $\varepsilon \to 0$. Here $\mathcal{A}_{\dag}^0$ is the effective operator with constant coefficients. For the norm of the difference of the operators $e^{-\mathcal{A}_{\dag ,\varepsilon}t}$ and $e^{-\mathcal{A}_{\dag}^0 t}$ a sharp order estimate (of order $O(\varepsilon)$) is obtained. Also, we find approximation for the exponential $e^{-\mathcal{A}_{\dag ,\varepsilon}t}$ in the $(L_2\rightarrow H^1)$-norm with error estimate of order $O(\varepsilon ^{1/2})$; in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.