Decay of waves in strain gradient porous elasticity with Moore–Gibson–Thompson dissipation

We study a one-dimensional problem arising in strain gradient porous-elasticity. Three different Moore–Gibson–Thompson dissipation mechanisms are considered: viscosity and hyperviscosity on the displacements, and weak viscoporosity. The existence and uniqueness of solutions are proved. The energy decay is also shown, being polynomial for the two first situations, unless a particular choice of the constitutive parameters is made in the hyperviscosity case. Finally, for the weak viscoporosity, only the slow decay can be expected. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.