On the time decay of solutions in porous-elasticity with quasi-static microvoids

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasi-static. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.     

Mindlin–Timoshenko systems with Kelvin–Voigt: analyticity and optimal decay rates

This paper is concerned with asymptotic stability of Mindlin–Timoshenko plates with dissipation of Kelvin–Voigt type on the equations for the rotation angles. We prove that the corresponding evolution semigroup is analytic if a viscoelastic damping is also effective over the equation for the transversal displacements. On the contrary, if the transversal displacement is undamped, we show that the semigroup is neither analytic nor exponentially stable. In addition, in the latter case, we show that the solution decays polynomially and we prove that the decay rate found is optimal.     

Stability of Timoshenko systems with past history

We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.     

Stabilisation faible interne locale de système élastique de Bresse

In [1], Alabau-Boussouira et al. (2011) studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. In this Note, our goal is to extend the results from Alabau-Boussouira et al. (2011) [1], by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.     

A transmission problem with non‐linear internal damping in elasticity

We consider an anisotropic body constituted by two different types of materials: a part is simple elastic while the other has a non‐linear internal damping. We show that the dissipation caused by the damped part is strong enough to produce uniform decay of the energy, more precisely, the energy decays exponentially when the dissipation is linear with respect to the velocity. For a non‐linear class of dissipations we prove that the energy decays polynomially. Copyright © 2003 John Wiley & Sons, Ltd.     

Radially symmetric solutions of the p-Laplacian in perforated-like domain with nonlocal boundary condition

In this paper, we study the stability of solutions to a von Kármán system for Kirchhoff plate equations with a memory condition working at the boundary. We show that such dissipation is strong enough to produce exponential decay of the solution provided the relaxation functions also decay exponentially. When the relaxation functions decay polynomially, we show that the solution decays polynomially.     

Exponential stability and partial averaging

The exponential stability of singularly perturbed time-varying systems is investigated. It turns out that, under natural conditions, exponential stability of an averaged system is equivalent to exponential stability of the perturbed system for small perturbation parameters. Explicit estimates for both, the approximation of single trajectories and the order of the exponential decay, are obtained. The method of proof does not require smoothness of the averaged system.     

Numerical analysis of a finite volume scheme for a seawater intrusion model with cross‐diffusion in an unconfined aquifer

We consider a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion. We propose and analyze a finite volume scheme based on two‐point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges toward a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.     

On the stability of explicit finite difference methods for advection–diffusion equations

In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.     

Stability conditions to Bresse systems with indefinite memory dissipation

ABSTRACT

We study the asymptotic behavior of Bresse system with non-dissipative kernel memory acting only in the equation of longitudinal displacement. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a new relationship between the coefficients of the system. Moreover, this new condition on the constants of the system is used to prove strong stability and exponential stability for the homogeneous case (frictional dissipation in the longitudinal equation).     

On pathwise super-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. In particular, we provide conditions for solutions to converge to zero at a given rate, which is faster than any exponential rate of decay. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.     

Stabilisation frontière indirecte du système de Timoshenko

In this Note, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. Under the equal speed wave propagation condition, we establish the exponential stability of the system. On the contrary, we show that the decay rate is polynomial.     

A boundary condition with memory in elasticity

In this paper, we study the stability of solutions of the n-dimensional nonhomogeneous and anisotropic elastic system with memory condition working at the boundary. We show that such dissipation is strong enough to produce exponential decay to the solution, provided the relaxation function also decays exponentially.     

Study of magnetosonic solitary waves for the electron magnetohydrodynamics equations

Electron magnetohydrodynamics equations are derived with allowance for nonlinearity, dispersion, and dissipation caused by friction between the ions and electrons. These equations are transformed into a form convenient for the construction of a numerical scheme. The interaction of codirectional and oppositely directed magnetosonic solitary waves with no dissipation is computed. In the first case, the solitary waves are found to behave as solitons (i.e., their amplitudes after the interaction remain the same), while, in the second case, waves are emitted that lead to decreased amplitudes. The decay of a solitary wave due to dissipation is computed. In the case of weak dissipation, the solution is similar to that of the Riemann problem with a structure combining a discontinuity and a solitary wave. The decay of a solitary wave due to dispersion is also computed, in which case the solution can also be interpreted as one with a discontinuity. The decay of a solitary wave caused by the combined effect of dissipation and dispersion is analyzed.     

Well-posedness and scattering of solutions to the sixth-order Boussinesq type equation in the framework of modulation spaces

In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local solutions and global solutions. Then scattering and stability of solutions are also discussed. The proof is mainly based on the decay properties of the solutions operator in modulation spaces and the contraction mapping principle.     

Stability and optimality of decay rate for a weakly dissipative Bresse system

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential decay in one-dimensional Type II/III thermoelasticity with two porosities

In this paper, we consider the theory of thermoelasticity with a double porosity structure in the context of the Green–Naghdi Types II and III heat conduction models. For the Type II, the problem is given by four hyperbolic equations, and it is conservative (there is no energy dissipation). We introduce in the system a couple of dissipation mechanisms in order to obtain the exponential decay of the solutions. To be precise, we introduce a pair of the following damping mechanisms: viscoelasticity, viscoporosities, and thermal dissipation. We prove that the system is exponentially stable in three different scenarios: viscoporosity in one structure jointly with thermal dissipation, viscoporosity in each structure, and viscoporosity in one structure jointly with viscoelasticity. However, if viscoelasticity and thermal dissipation are considered together, undamped solutions can be obtained     

Stability and persistence of two-dimensional patterns

The canonical equations for evolution of the amplitude order parameters order parameters describing the nonlinear development and persistence of two-dimensional three-mode spatial patterns generated by Turing instability in dissipative systems are considered. The stability conditions for persistent hexagonal patterns are generalized, and the conditions under which patterns are either disrupted, exhibit bounded quasiperiodic or chaotic behavior, or decay under nonlinear evolution are derived. These conditions are applied to the specific three-mode amplitude evolution equations derived for the Schnakenberg model and a delay predator system in Chapter 3. Numerical results are presented for the persistence, disruption and decay of patterns in these systems, including fairly detailed comparisons with simulations results for the Snackenberg model.     

Energy stability for a class of two-dimensional interface linear parabolic problems

We consider parabolic equations in two-dimensions with interfaces corresponding to concentrated heat capacity and singular own source. We give an analysis for energy stability of the solutions based on special Sobolev spaces (the energies also are given by the norms of these spaces) that are intrinsic to such problems. In order to define these spaces we study nonstandard spectral problems in which the eigenvalue appears in the interfaces (conjugation conditions) or at the boundary of the spatial domain. The introducing of appropriate spectral problems enable us to precise the values of the parameters which control the energy decay. In fact, in order for numerical calculation to be carried out effectively for large time, we need to know quantitatively this decay property.     

SOME STABILITY RESULTS FOR TIMOSHENKO SYSTEMS WITH COOPERATIVE FRICTIONAL AND INFINITE-MEMORY DAMPINGS IN THE DISPLACEMENT

In this paper, we consider a vibrating system of Timoshenko-type in a onedimensional bounded domain with complementary frictional damping and infinite memory acting on the transversal displacement. We show that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case. We establish in each case a general decay estimate of the solutions. In the particular case when the wave propagation speeds are different and the frictional damping is linear, we give a relationship between the smoothness of the initial data and the decay rate of the solutions. By the end of the paper, we discuss some applications to other Timoshenko-type systems.