Stability of Abstract Linear Thermoelastic Systems With Memory

Abstract

An abstract linear thermoelastic system with memory is here considered. Existence, uniqueness, and continuous dependence results are given. In presence of regular and convex memory kernels, the system is shown to be exponentially stable. An application to the Kirchhoff plate equation is given.

On the energy decay of the linear thermoelastic plate with memory

“…Following the ideas of [1], this requires the introduction of an additional variable, namely, the summed past history of ϑ (cf. [3]), defined as…”

“…For the sake of simplicity, we put all the other physical constants equal to one. Actually, in the original setting proposed by Lagnese [6] and developed in [3] for the present model, one has c > 0. Nonetheless, there are other models in the literature which take c = 0 (cf., for instance, [2]).…”

“…A similar problem was considered in [3], although with different boundary conditions (namely, assuming that the plate is supported and ϑ satisfies the homogeneous Dirichlet boundary condition). There, the main difference with respect to our model is the presence of a (dissipative) term ϑ in the second equation, arising from the assumption that, besides the heat flux, also the thermal power depends on the past history of ϑ .…”

“…Then, reformulating the problem in terms of C 0 -semigroups on Hilbert spaces accounting for the past history of ϑ , and exploiting techniques from abstract semigroup theory (see, e.g., [7]), the exponential stability was demonstrated, under very weak assumptions on the memory kernel. Moreover, the authors conjectured that in absence of memory effects in the thermal power constitutive law (so that the extra term ϑ disappears) the exponential stability may fail (see [3,Remark 5.1]). This is exactly what we prove in this work.…”

Long-time behavior of a class of thermoelastic plates with nonlinear strain

“…Then, without p-Laplacian, problem (1.2)-(1.3) models non-Fourier thermoelastic plates (cf. [17,Sec. 5]).…”