ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND BLOWUP OF THE SOLUTION

In this paper we consider the Elastic membrane equation with memory term and nonlinear boundary damping.Under some appropriate assumptions on the relaxation function h and with certain initial data,the global existence of solutions and a general decay for the energy are established using the multiplier technique.Also,we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.     

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.     

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.     

Exponential decay for a viscoelastically damped timoshenko beam

Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-known that the system is exponentially stable if the kernel in the memory term is sub-exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions: What if the kernel is tested against a different function (say Gamma) other than the exponential function? Would there still be stability? In the affirmative, what kind of decay rate we get? It is proved that for a non-decreasing function “Gamma” whose “logarithmic derivative” is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.     

Non-existence for fractionally damped fractional differential problems

In this paper, we are concerned with a fractional differential inequality containing a lower order fractional derivative and a polynomial source term in the right hand side. A non-existence of non-trivial global solutions result is proved in an appropriate space by means of the test-function method. The range of blow up is found to depend only on the lower order derivative. This is in line with the well-known fact for an internally weakly damped wave equation that solutions will converge to solutions of the parabolic part.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.     

On a large class of kernels yielding exponential stability in viscoelasticity

In this paper we prove exponential decay of solutions for a problem which appears in viscoelasticity. The conditions on the admissible kernels are relaxed so as to allow for more kernels to be treated. Namely, the smallness of the kernels is replaced by the smallness of the set where the kernel is flat. This work extends previous works and improves in particular a recent result by Pata [16]. This is established by introducing two lemmas, an idea due to Pata and using the energy method.