Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

<p style='text-indent:20px;'>In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.</p>

A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511–541; F. Alabau-Boussouira and M. Léautaud, ESAIM: COCV 18 (2012) 548–582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643–669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61–105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473–1517].

Symmetries and conservation laws of a damped Boussinesq equation

In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

A PEAK IN THE BOSE GLASS LINE AT THE FIRST MATCHING FIELD OF NANOSTRUCTURES OF PINNING SITES

Using an extensive series of molecular dynamic simulations on driven vortex lattices interacting with nanostructures of periodic square arrays of pinning sites of fixed density, we have obtained the B–T phase diagram representing a Bose glass to vortex liquid-phase transition. In solving the over damped equation of vortex motion, we took into account the vortex–vortex repulsion interaction, the attractive vortex–pinning interaction, and the driving Lorentz force at several values of temperature. We have found that the location of the Bose glass line (TBG-line) depends on the pinning strength of the pinning sites. We have also found that for each pinning strength value, there is an interesting peak of the TBG-line that occurs at the first matching field when the number of vortices equals the number of pinning sites in the sample.

Convergence of the strongly damped nonlinear Klein-Gordon equation in ℝn with radial symmetry

SynopsisWe consider the strongly-damped Klein–Gordon equation in ℝ3 in the case where the initial data possess radial symmetry. With the latter assumption we are able to extend the result of [2] which assumed a bounded spatial domain. Specifically, we construct a global weak solution v of theundamped equation for high powers p which can be approximated arbitrarily closely (for small α) by the global strong solutions of the damped equation found by Aviles and Sandefur [1].