On the Homogenization of a Parabolic Equation with a Nonlocal Convolution Type Operator

Abstract

In \(L_2(\mathbb{R}^d)\), we consider a self-adjoint bounded operator \({\mathbb A}_\varepsilon(t)\), \(t\ge 0\), \(\varepsilon >0\), of the form

It is assumed that \(a(\cdot)\in L_1(\mathbb R^d)\) is a nonnegative function such that \(\int_{\mathbb{R}^d} | \mathbf{x} |^3 a(\mathbf{x})\,d\mathbf{x}<\infty\) and \(a(-\mathbf{x}) = a(\mathbf{x})\); the function \(\mu(\mathbf{x},\mathbf{y},t)\) is measurable and periodic in all variables, moreover, \(\mu(\mathbf{x},\mathbf{y},t) = \mu(\mathbf{y},\mathbf{x},t)\) and \(0< \mu_- \leqslant \mu(\mathbf{x},\mathbf{y},t) \leqslant \mu_+< \infty\). Let \(u_\varepsilon(\mathbf{x},t)\), \(\mathbf{x} \in {\mathbb R}^d\), \(t \ge 0\), be the solution of the Cauchy problem

where \(\varphi \in L_2({\mathbb R}^d)\). We show that, for a chosen \(t>0\), the solution \(u_\varepsilon(\cdot,t)\) converges in \(L_2({\mathbb R}^d)\), as \(\varepsilon \to 0\), to the solution \(u_0(\cdot,t)\) of the homogenized problem

where \({\mathbb A}^0= - \operatorname{div} g^0 \nabla\) and \(g^0\) is a positive definite effective matrix. The following error estimate holds: