Timoshenko systems with fading memory

The decay properties of the semigroup generated by a linear Timoshenko system with fading memory are discussed. Uniform stability is shown to occur within a necessary and sufficient condition on the memory kernel

Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type $$\begin{cases} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\ \rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) - \delta A^\gamma {\theta} = 0\\ \rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{cases}$$ where the operator $A$ is strictly positive selfadjoint. For any fixed $\gamma\in\mathbb{R}$, the stability properties of the related solution semigroup $S(t)$ are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of $S(t)$ when the spectrum of the leading operator $A$ is not made by eigenvalues only.Comment: Corrected typo

On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction

We 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.Comment: Abstract shortened and few typos correcte

A quantitative Riemann-Lebesgue lemma with application to equations with memory

An elementary proof of a quantitative version of the Riemann-Lebesgue lemma for functions supported on the half line is given. Applications to differential models with memory are discussed

Steady states of elastically-coupled extensible double-beam systems

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases}$$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations

A short elementary proof of the Gearhart-Pr\"uss theorem for bounded semigroups

We present a short elementary proof of the Gearhart-Pr\"uss theorem for bounded $C_0$-semigroups on Hilbert spaces.Comment: Accepted for publication in the Proceedings of the conference 'Control & Inverse Problems' (Monastir, Tunisia, May 2022), Springer Natur

Ionic dialysance allows an adequate estimate of urea distribution volume in hemodialysis patients

Ionic dialysance allows an adequate estimate of urea distribution volume in hemodialysis patients.BackgroundAn adequate estimation of urea distribution volume (V) in hemodialysis patients is useful to monitor protein nutrition. Direct dialysis quantification (DDQ) is the gold standard for determining V, but it is impractical for routine use because it requires equilibrated postdialysis plasma water urea concentration. The single pool variable volume urea kinetic model (SPVV-UKM), recommended as a standard by Kidney Disease Outcomes Quality Initiative (K/DOQI), does not need a delayed postdialysis blood sample but it requires a correct estimate of dialyser urea clearance.MethodsIonic dialysance (ID) may accurately estimate dialyzer urea clearance corrected for total recirculation. Using ID as input to SPVV-UKM, correct V values are expected when end-dialysis plasma water urea concentrations are determined in the end-of-session blood sample taken with the blood pump speed reduced to 50 mL/min for two minutes (Upwt2′). The aim of this study was to determine whether the V values determined by means of SPVV-UKM, ID, and Upwt2′ (VID) are similar to those determined by the “gold standard” DDQ method (VDDQ). Eighty-two anuric hemodialysis patients were studied.ResultsVDDQ was 26.3 ± 5.2 L; VID was 26.5 ± 4.8 L. The (VID–VDDQ) difference was 0.2 ± 1.6 L, which is not statistically significant (P = 0.242). Anthropometric volume (VA) calculated using Watson equations was 33.6 ± 6.0 L. The (VA–VDDQ) difference was 7.3 ± 3.3 L, which is statistically significant (P < 0.001).ConclusionAnthropometric-based V values overestimate urea distribution volume calculated by DDQ and SPVV-UKM. ID allows adequate V values to be determined, and circumvents the problem of delayed postdialysis blood samples