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.     

Global solvability and decay estimates for a type III thermo-viscoelastic coupled system with infinite memory and boundary interaction feedback

A type III thermo-viscoelastic coupled system with infinite memory and distributed delay is considered. The interaction feedback between the nonlinear damping and the acoustic conditions are reacted on portion of the boundary. We obtain the well posedness and regularity of the system by using semigroup theory which is combined with Schauder's fixed point theorem. Moreover, the general decay estimates are established under a much larger class of relaxation functions. Our results are obtained without the boundedness condition of initial data assumed in many earlier papers in the literature. This work generalizes the composite stability between infinite memory and nonlinear damping.     

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.     

Exponential decay for a von Kármán equations of memory type with acoustic boundary conditions

In this paper, we consider the dynamical von Kármán equations of memory type with acoustic boundary conditions. We show an exponential decay result of solutions under weaker assumption than the ones frequently used in the literature. In particular, the kernel we are considering is not necessarily exponentially decaying to zero as was assumed before. Copyright © 2014 John Wiley & Sons, Ltd.     

General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with relation between the damping term and relaxation function. The second result is given without imposing any restrictive growth assumption on the damping term and the kernel function \(g\). Our result extends the decay result obtained for problems with finite history to those with infinite history.     

Quasi-optimized Schwarz methods for reaction diffusion equations with time delay

In this paper, we investigate the convergence behavior of the Schwarz waveform relaxation (SWR) algorithms for solving PDEs with time delay. We choose the reaction diffusion equations with a constant time delay as the underlying model problem and try to derive optimized transmission conditions of Robin type. To this end, we propose a new method to get quasi-optimized parameter involved in the transmission conditions and it is shown that this method is essentially different from the existing ones. Moreover, when the situation is reduced into the heat equations with a constant delay, we show that this method results in a more efficient quasi-optimized parameter. Numerical results are provided to validate our theoretical results.     

General decay for a von Karman equation of memory type with acoustic boundary conditions

A Karman equation of memory type with acoustic boundary conditions is considered. This work is devoted to investigate the influence of kernel function g and prove general decay rates of solutions when g does not necessarily decay exponentially.     

Nonlinear character dependent models with constant time delay in population dynamics

A nonlinear character dependent model with constant time delay that extends the linear model of W. S. C. Gurney and R. M. Nisbet is introduced as a result of making the birth and death moduli dependent upon the total population. We reduce this model—an initial boundary value problem—to the solution of a pair of coupled nonlinear functional equations. Under suitable conditions, we establish the local existence, uniqueness, and continuation of solutions for all positive time. We also establish necessary and sufficient conditions for the local stability of time independent solutions in the form of a Lotka type characteristic equation similar to M. E. Gurtin and R. C. MacCamy's charateristic equation arising from their nonlinear extension of the age dependent MacKendrick-Von Foerster model.     

Asymptotic behavior to a von Kármán equations of memory type with acoustic boundary conditions

We study the stability of solutions to a von Kármán plate model of memory type with acoustic boundary conditions. We establish the general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve on some earlier results-exponential or polynomial decay rates.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

General Energy Decay for a Viscoelastic Equation of Kirchhoff Type with Acoustic Boundary Conditions

This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of \(\mathbb {R}^{n}.\) We show that, under suitable conditions on the initial data, the solution exists globally in time. Then, we prove the general energy decay of global solutions by applying a lemma of Martinez, which allows us to get our decay result for a class of relaxation functions wider than that usually considered.     

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

In this paper, we are concerned with a one‐dimensional porous‐thermoelastic system of type III with a viscoelastic damping and boundary time‐varying delay. Under suitable assumptions on relaxation function and time delay, we establish the exponential decay result of the system in which the damping is strong enough to stabilize the thermoelastic system in the presence of time delay. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability for the damped wave equation with neutral delay

Of concern is a wave equation with a distributed neutral delay. We prove that, despite the destructive nature of delays in general, solutions may go back to the equilibrium state in an exponential manner as time goes to infinity. Reasonable conditions on the distributed neutral delay are established. This type of problems appear in the study of wave propagation in viscoelastic media and in acoustic wave propagation. It is not well studied so far.     

General decay of solutions of a wave equation with memory term and acoustic boundary condition

In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non‐locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.     

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u≡1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain.     

On a system of Klein-Gordon type equations with acoustic boundary conditions

We prove the existence, uniqueness and uniform stabilization of global solutions for a generalized system of Klein-Gordon type equations with acoustic boundary conditions on a portion of the boundary and the Dirichlet boundary condition on the rest.     

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.     

Dynamic properties for nonlinear viscoelastic Kirchhoff-type equation with acoustic control boundary conditions II

In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation with initial conditions and acoustic boundary conditions. Under suitable conditions on the initial data, the relaxation function $h(\cdot)$ and $M(\cdot)$, we prove that the solution blows up in finite time and give the upper bound of the blow-up time $T^*$.     

GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.