Asymptotical stability for memory‐type porous thermoelastic system of type III with constant time delay

In this paper, we consider a one‐dimensional porous thermoelastic system of type III with viscoelastic damping and constant time delay on boundary. Using the energy method, we prove the general stability of the system under suitable assumptions on the relaxation function and the time delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilization of a type III thermoelastic Timoshenko system in the presence of a time‐distributed delay

In this paper, we investigate a Timoshenko‐type system of thermoelasticity of type III in the presence of a distributed delay. We prove the well‐posedness and two exponential stability results in the presence as well as in the absence of an extra frictional damping. In case of absence of the frictional damping, our stability result is obtained under the equal‐speed of propagation and a smallness condition on the weight of delay.     

Stability of Timoshenko systems with thermal coupling on the bending moment

The Timoshenko system is a distinguished coupled pair of differential equations arising in mathematical elasticity. In the case of constant coefficients, if a damping is added in only one of its equations, it is well‐known that exponential stability holds if and only if the wave speeds of both equations are equal. In the present paper we study both non‐homogeneous and homogeneous thermoelastic problems where the model's coefficients are non‐constant and constants, respectively. Our main stability results are proved by means of a unified approach that combines local estimates of the resolvent equation in the semigroup framework with a recent control‐observability analysis for static systems. Therefore, our results complement all those on the linear case provided in [22], by extending the methodology employed in [4] to the case of Timoshenko systems with thermal coupling on the bending moment.     

A uniform stability result for thermoelasticity of type III with boundary distributed delay

In this paper we consider a thermoelastic system of type III with boundary distributed delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Green and Naghdi's theory is still strong enough to uniformly stabilize the system even in the presence of time delay.     

Asymptotic behavior of second sound thermoelasticity with internal time-varying delay

In this paper, we consider a thermoelastic system of second sound with internal time-varying delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Cattaneo’s law is still strong enough to uniformly stabilize the system even in the presence of time delay.     

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem. A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔu_tt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of u_t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u_t we use analyticity of the corresponding thermoelastic semigroup proved in [25]. The isothermal variant of this problem with additional mechanical damping term −εΔu_t , ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain.     

Uniform stability for thermoelastic systems with boundary time-varying delay

In this paper we consider a thermoelastic system with boundary time-varying delay. Using the energy method, we show, under suitable assumptions, that the damping effect through heat conduction is still strong enough to uniformly stabilize the system even in the presence of boundary time-varying delay. Our result improves earlier results existing in the literature.     

General decay of solutions of quasilinear wave equation with time‐varying delay in the boundary feedback and acoustic boundary conditions

In this paper, we are concerned with the general decay result of the quasi‐linear wave equation with a time‐varying delay in the boundary feedback and acoustic boundary conditions. Copyright © 2017 John Wiley & Sons, Ltd.     

A classification of structural dissipations for evolution operators

In this paper, we study the asymptotic profile of the solution for a σ‐evolution equation with a time‐dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley & Sons, Ltd.     

On stability of hyperbolic thermoelastic Reissner–Mindlin–Timoshenko plates

In the present article, we consider a thermoelastic plate of Reissner–Mindlin–Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absence of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, and so on. We present a well‐posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending component is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovski? operator for irrotational vector fields, which we discuss in the appendix. Copyright © 2014 John Wiley & Sons, Ltd.     

Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect

In this paper, we consider travelling wave solutions for the diffusive Nicholson’s blowflies equation incorporating time delay and diffusion. Special attention is paid to the modelling of the time delay to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For the strong generic delay kernel, we show that travelling wave solutions exist provided that the delay is sufficiently small, using the geometric singular perturbation theory.     

Unilateral problems for the wave equation with degenerate and localized nonlinear damping: well‐posedness and non‐stability results

Unilateral problems related to the wave model subject to degenerate and localized nonlinear damping on a compact Riemannian manifold are considered. Our results are new and concern two main issues: (a) to prove the global well‐posedness of the variational problem; (b) to establish that the corresponding energy functional is not (uniformly) stable to equilibrium in general, namely, the energy does not converge to zero on the trajectory of every solution, even if a full linear damping is taken in place.     

Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one‐dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.     

Multiple‐interval‐dependent robust stability analysis for uncertain stochastic neural networks with mixed‐delays

This article deals with the problem of robust stochastic asymptotic stability for a class of uncertain stochastic neural networks with distributed delay and multiple time‐varying delays. It is noted that the reciprocally convex approach has been intensively used in stability analysis for time‐delay systems in the past few years. We will extend the approach from deterministic time‐delay systems to stochastic time‐delay systems. And based on the new technique dealing with matrix cross‐product and multiple‐interval‐dependent Lyapunov–Krasovskii functional, some novel delay‐dependent stability criteria with less conservatism and less decision variables for the addressed system are derived in terms of linear matrix inequalities. At last, several numerical examples are given to show the effectiveness of the results. © 2014 Wiley Periodicals, Inc. Complexity 21: 147–162, 2015     

Global solutions to the Navier‐Stokes‐Landau‐Lifshitz system

This paper is dedicated to the study of the Navier‐Stokes‐Landau‐Lifshitz system. We obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field. Our methods mainly rely upon the Fourier frequency localization and Bony's paraproduct decomposition.     

Existence of solutions for a class of second‐order differential equations with impulsive effects

In this paper, by using the variational method and the critical point theorem because of Brezis and Nirenberg, we investigate the existence of solutions to a class of second‐order impulsive Hamiltonian system. Copyright © 2015 John Wiley & Sons, Ltd.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

We establish the existence of local in time semi‐strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness of local semi‐strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.