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Global attractors for strongly damped wave equations with subcritical-critical nonlinearities
No full textOn the stability of Bresse and Timoshenko systems with hyperbolic heat conduction
Full text linkWe investigate the stability of three thermoelastic beam systems with hyperbolic heat conduction. First, we study the Bresse-Gurtin-Pipkin system, providing a necessary and sufficient condition for the exponential stability and the optimal polynomial decay rate when the condition is violated. Second, we obtain analogous results for the Bresse-Maxwell-Cattaneo system, completing an analysis recently initiated in the literature. Finally, we consider the Timoshenko-Gurtin-Pipkin system and we find the optimal polynomial decay rate when the known exponential stability condition does not hold. As a byproduct, we fully recover the stability characterization of the Timoshenko-Maxwell-Cattaneo system. The classical “equal wave speeds” conditions are also recovered through singular limit procedures. Our conditions are compatible with some physical constraints on the coefficients as the positivity of the Poisson's ratio of the material. The analysis faces several challenges connected with the thermal damping, whose resolution rests on recently developed mathematical tools such as quantitative Riemann-Lebesgue lemmas
On the energy inequality for weak solutions to the Navier-Stokes equations of compressible fluids on unbounded domains
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On the Moore-Gibson-Thompson equation with thermal effects of Gurtin-Pipkin type
No full textWe 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
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