This work deals with a class of evolution problems consisting of a pair of coupled equations for modelling propagation of elastic waves in fluid-saturated porous media. The type of the first equation depends on two physical parameters (density and secondary consolidation) which can vanish while the second one is always parabolic. In case the density never vanishes, the first equation is second-order hyperbolic type and a weak solution to the problem is constructed using a variational method in a Sobolev framework. Next, the proof of uniqueness involves Ladyzenskaja's test-functions used to compensate a lack of regularity that would be required in a standard energy method. This approach gives rise to a priori estimates which are useful to prove that the linearized thermoelasticity and the quasi-static systems are defined as asymptotic models of the Biot problem when the secondary consolidation coefficient or the density is small.
"Asymptotic Biot's models in porous media." Adv. Differential Equations 11 (1) 61 - 90, 2006.