: In this thesis, we studied the existence, uniqueness and the asymptotic behavior of some systems with two porous structures in one dimensional spatial space.
The first system is an elastic system with double porosity structure, we have shown the existence, uniqueness and the exponential stability of the solution in two cases of the viscoelastic dissipation.
The second system is an elastic system with two porous structures and memory effects in both porous equations. We prove that the weak dissipation generated by the memory terms produces a general rate of decay depending on the kernels of the memory terms and the coefficients of the system.
The methods we have used are the spectral method based on the semigroup theory and Gerhart Pruss theorem and the multiplier method based on the energy estimate and Lyapunov direct method.