Nguetseng's Two-scale Convergence Method For Filtration and Seismic Acoustic Problems in Elastic Porous Media

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results, we derive Biot's equations of poroelasticity, equations of viscoelasticity, or decoupled system consisting of non-isotropic Lam\'{e}'s equations and Darcy's system of filtration, depending on ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures

Asymptotic behavior of a semilinear problem in heat conduction with memory

3This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.openopenC. GIORGI; MARZOCCHI A.; PATA V.Giorgi, Claudio; Marzocchi, Alfredo; Pata, Vittorin