Well-posedness and exponential stability in nonlocal theory of nonsimple porous thermoelasticity

In this paper, a nonlocal model for porous thermoelasticity is derived in the framework of the Mindlin’s strain gradient theory where the thermal behavior is based on the entropy balance postulated by Green–Naghdi models of type II or III. In this context, we add the second gradient of volume fraction field and the second gradient of deformation to the set of independent constituent variables. The elastic nonlocal parameter \(\varpi\) and the strain gradient length scale parameter l are also considered in the derived model. New mathematical difficulties then appeared due to the higher gradient terms in the resulting system which is a coupling of three second order equations in time. By using the theories of monotone operators and the nonlinear semigroups, we prove the well-posedness of the derived model in the one dimensional setting. The exponential stability of the corresponding semigroup to type II and type III models is proved. The proof is essentially based on a characterization stated in the book of Liu and Zheng. This result of exponential stability of the type II model confirms the results of the classic theory for which the exponential decay cannot hold (for type II) without adding a dissipative mechanism.