On the spatial behavior of solutions for porous elastic solids with quasi-static microvoids

The spatial behavior of solutions for the dynamic problem of elasticity with voids is well known. It has been analyzed making use of the usual techniques for hyperbolic differential equations. Here we consider a related but different problem, the problem determined by the dynamic deformations of porous elasticity with porous dissipation when the motion of the microvoids is supposed to be quasi-static. We think that this kind of dynamical problem deserves some consideration. However, little attention has been paid to it in the literature. On the one hand, the problem has interest from a mechanical point of view, because this is the system that one must study when the microvoids deformations are small enough as to assume that their acceleration is nearly zero. On the other hand, the problem has also interest from its mathematical structure, because the coupling of a parabolic equation with a hyperbolic one arises in the system.