Lack of differentiability in nonlocal nonsimple porous thermoelasticity with dual-phase-lag law

In this paper we consider the nonlocal theory for porous thermoelastic materials based on Mindlin’s strain gradient theory with nonlocal dual-phase-lag law. This makes the derived equations more physically realistic, as they overcome the infinite propagation velocity property of the Fourier law. This approach consists of adding the second strain gradient and the second volume fraction gradient field to the set of independent constituent variables. We then obtain a system of three second order time equations with higher gradient terms. Using semigroup theory, we show the well-posedness of the one-dimensional problem. By an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we prove that the associated semigroup is exponentially stable but not differentiable. The lack of analyticity and the impossibility to localize the solutions in time are direct consequences.