We consider the Moore-Gibson-Thompson-Gurtin-Pipkin model $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\ \noalign{\vskip1mm} \displaystyle \theta_t - \int_0^\infty g(s)\Delta\theta(t-s)ds = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t \end{cases} $$ with the first equation in the subcritical regime $\alpha\beta>\gamma$. 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 $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\ \noalign{\vskip1mm} \displaystyle \theta_t - \nu\Delta\theta = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t \end{cases} $$ 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
Dell'Oro, Filippo;Pata, Vittorino
2023-01-01
Abstract
We consider the Moore-Gibson-Thompson-Gurtin-Pipkin model $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\ \noalign{\vskip1mm} \displaystyle \theta_t - \int_0^\infty g(s)\Delta\theta(t-s)ds = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t \end{cases} $$ with the first equation in the subcritical regime $\alpha\beta>\gamma$. 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 $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\ \noalign{\vskip1mm} \displaystyle \theta_t - \nu\Delta\theta = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t \end{cases} $$ formally obtained as a limit by letting $g\to\nu\delta_{0^+}$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
