The focus of this work is to investigate the well-posedness and exponential stability of a swelling porous system with the Gurtin–Pipkin thermal effect as the only source of damping. The well-posedness result is achieved using an essential corollary to the Lumer–Phillips Theorem. By constructing a suitable Lyapunov functional, we establish an exponential stability result without the conventional limitation to the system’s parameters (coined a stability number in the literature). Generally, the study demonstrates that the unique dissipation from the Gurtin–Pipkin thermal law is sufficient to stabilize the system exponentially, irrespective of the system’s parameters.

/pdf/well-posedness-and-exponential-stability-of-swelling-porous-3py78wki.pdf

Well-Posedness and Exponential Stability of Swelling Porous with Gurtin–Pipkin Thermoelasticity

In the present paper we consider the Standard Linear Solid model in R N coupled with the Cattaneo law of heat conduction. We show the well-posedness and asymptotic stability of the problem, giving decay rates for a norm related to the solution. These results are compared with those given for the Fourier problem in (Pellicer and Said-Houari (2020)) and the ones of the problem without heat conduction (see previous work (Appl Math. Optim 80 (2019) 447–478)). The main difference is that the Cattaneo system exhibits the well-known regularity-loss phenomenon. The methods used to prove these results are the energy method in the Fourier space and the eigenvalues expansion method.

On the Cauchy problem of the standard linear solid model with Cattaneo heat conduction

Optimal decay rate for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law

We consider the Moore-Gibson-Thompson-Gurtin-Pipkin model \begin{document}$ \left\{ \begin{array}{l}u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u = - \varrho\Delta \theta \\ \begin{align} \theta_t - \int_0^\infty g(s)\Delta\theta(t-s)ds = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t\end{align} \end{array} \right. $\end{document} with the first equation in the subcritical regime $ \alpha\beta&gt;\gamma $. The system generates a strongly continuous semigroup of linear contractions which is never exponentially stable, even if the second equation, when uncoupled, generates an exponentially stable semigroup. This is in deep contrast to what happens in connection with the semigroup generated by the Moore-Gibson-Thompson-Fourier system \begin{document}$ \left\{ \begin{array}{l}u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u = - \varrho\Delta \theta \\ \theta_t - \nu\Delta\theta = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t \end{array} \right. $\end{document} formally obtained as a limit by letting $ g\to\nu\delta_{0^+} $. 

On the Moore-Gibson-Thompson equation with thermal effects of Gurtin-Pipkin type

/pdf/decay-properties-for-the-cauchy-problem-of-the-linear-jmgt-318oubdr.pdf

Decay properties for the Cauchy problem of the linear JMGT-viscoelastic plate with heat conduction

In this paper, we consider the Cauchy problem related to the standard linear solid model with Gurtin–Pipkin thermal law in the whole space. Under some assumptions on the relaxation function g, we establish the well-posedness result by using semigroup theory. Besides, by using the energy method in the Fourier space, we prove the decay estimate result under the non-critical case. Our result indicates that the decay property is of the regularity-loss type, which is in line with the decay property of Cattaneo system.

Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law

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Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law