Decay property of regularity-loss type for solutions in elastic solids with voids

Polynomial stability for the equations of porous elasticity in one-dimensional bounded domains

“…To the best of the authors' knowledge, the first papers dealing with the regularity loss property for solutions of porous-elastic systems are due to Said-Houari and Messaoudi Messaoudi [11] and Djouamaia and Said-Houari [12]. In [11], Said-Houari and Messaoudi considered a porous-elastic system in the scenario of the non-classical thermo-elasticity given by…”

“…(1.15) τ 0 q t + q + kθ x = 0, x ∈ R, t > 0, (1.16) and they showed that the solutions decay slowly being of regularity loss type. On the other hand, Djouamaia and Said-Houari [12] considered the porous-elastic system given by…”

Decay Structures for the Equations of Porous Elasticity in One-Dimensional Whole Space

“…It is worth noting that this kind of problem has been studied in a deep way in the case of bounded domains [6,8,9,10,11,12]. However, the case when the domain is the whole space has been very forgotten in the literature and as far as we know we can only recall the contributions [2,14]. Unfortunately, the decay estimates introduced in [2] are not optimal and they considered only the dissipation mechanisms at the elastic structure.…”

“…However, the case when the domain is the whole space has been very forgotten in the literature and as far as we know we can only recall the contributions [2,14]. Unfortunately, the decay estimates introduced in [2] are not optimal and they considered only the dissipation mechanisms at the elastic structure. In [14] the authors studied the porous elastic problem when viscosity and non-classical thermal effects were present and they obtained results concerning slow decay and regularity-loss type.…”

Decay Rates of the Solutions to the Thermoelastic Bresse System of Types I and III

“…Be that as it may, from mathematical point of view, a considerable number of stability issues concerning the Bresse model in a whole space, have received considerable attention in the last years [10,24,25,28,31]. This has been due to the regularity-loss phenomenon that usually appears in the pure Cauchy problems (for instance, see [5,6,12,13,14,32] and references therein). Roughly speaking, the decay rate of the solution is of the regularity-loss type, when it is obtained only by assuming some additional order regularity on the initial date.…”