Exponential decay for a neutral wave equation

A strongly damped wave equation involving a delay of neutral type in its second order derivative is considered. It is proved that solutions decay to zero exponentially despite the fact that delays are, in general, sources of instability.     

Stability of a thermoelastic laminated system subject to a neutral delay

In this work, we consider a system of two identical beams of uniform thickness modeled as a Timoshenko system. The slip between the beams is taken into account, and the system is coupled with a heat equation. Moreover, the slip equation is subject to a distributed delay of neutral type. Delays are known to be of a destructive nature in general. Therefore, here, the delay will compete with a frictional damping and the dissipation produced by the heat equation. We provide sufficient conditions ensuring exponential and polynomial stability of the structure.     

Stability of a critical nonlinear neutral delay differential equation

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of constant and periodic solutions. A complete analysis of the stability diagram is given in the linear homogeneous case. Under periodic forcing, existence of periodic solutions is discussed, involving a Diophantine condition on the period of the source.     

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation

(1)     

A blow up result for a fractionally damped wave equation

In this paper we prove a blow up result for solutions of the wave equation with damping of fractional order and in presence of a polynomial source. This result improves a previous result in [5]. There we showed that the classical energy is unbounded provided that the initial data are large enough.     

Existence of solutions for impulsive partial neutral functional differential equation with infinite delay

In this paper, the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces is investigated. We derive conditions in respect of the Hausdorff measure of noncompactness under which the mild solutions exist in Banach spaces. Our results improve and generalize some previous results.     

Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation

_utt−Δ_ut−Δu+f(_ut)+g(u)=h$u_{tt}-\Delta u_t-\Delta u+f\left(u_t\right)+g\left(u\right)=h$

with Dirichlet boundary conditions, where both the nonlinearities f$f$ and g$g$ exhibit a critical growth, while h$h$ is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.     

Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

We consider the Cauchy problem for the damped wave equation with absorption

(∗)     

Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

Polynomial decay rate for the dissipative wave equation

Using Fourier integral operators with special amplitude functions, we analyze the stabilization of the wave equation in a three-dimensional bounded domain on which exists a trapped ray bouncing up and down infinitely between two parallel parts of the boundary.     

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.     

Existence results for an impulsive neutral functional differential equation with state-dependent delay

In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray–Schauder Alternative fixed point theorem.     

Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case

In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson's equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities.     

Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1/(ln(t+2))^2−δ,∀δ>0, when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1/(ln(ln(t+e²)))^2−δ,∀δ>0.     

Critical exponent for damped wave equations with nonlinear memory

We consider the Cauchy problem in Rⁿ,n≥1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t→∞ of small data solutions have been established in the case when 1≤n≤3. We also derive a blow-up result under some positive data in any dimensional space.     

Periodic solutions for generalized high-order neutral differential equation in the critical case

By applying Mawhin’s continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for a generalized high-order neutral differential equation in the critical case. Moreover, an example is given to illustrate the results.     

Some results on second-order neutral functional differential equations with infinite distributed delay

In this paper, we consider two types of second-order neutral functional differential equations with infinite distributed delay. By choosing available operators and applying Krasnoselskii’s fixed-point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations.     

Large time behaviors of hot spots for the heat equation with a potential

We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.