Solvability of equations with convolutions that arise in homogenization problems
Keywords:initial-boundary-value problem, homogenized equation, a priori estimate, Laplace transform
We consider the initial-boundary-value problems for the non-stationary equations of filtration in porous media. Such problems are relevant in the underground water pollution control. We consider the periodic media with a small microscale coefficient as models of porous media. We present the solvability and regularity theorems for the corresponding homogenized problems with convolutions. These theorems are formulated for general input data and non-homogeneous initial conditions, and they extend the classical solvability theorems for the heat equation. To prove the theorems, we use the a priori estimate method and the well-known Agranovich—Vishik method.
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