Nasser-eddine Tatar

Existence and uniqueness for a problem involving Hilfer fractional derivative

We consider an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. We prove existence and uniqueness of global solutions in the space of weighted continuous functions. The stability of the solution for a weighted Cauchy-type problem is also analyzed.

Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel

In this paper we consider a problem which arises in viscoelasticity. We prove an exponential decay of solutions under weaker assumptions than the ones frequently used in the literature. In particular, the kernels we are considering are not necessarily exponentially decaying to zero as was assumed before. The present results improve also a previous work of the authors.

Arbitrary decays in linear viscoelasticity

It is by now well known that a necessary condition for the exponential (polynomial) decay of the energy of a problem arising in viscoelasticity is that the kernel (which appears in the memory term) itself be of exponential (polynomial) type. By a kernel of exponential (polynomial) type we mean that the product of this kernel with an exponential (polynomial) function is summable. Some researchers have started from this condition to seek other (sufficient) conditions ensuring exponential or polynomial decay of the energy. In this work we generalize these works to an arbitrary decay not necessarily of an exponential or polynomial rate.

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.

An impulsive nonlinear singular version of the Gronwall-Bihari inequality

We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.

Uniform stabilization of solutions of a nonlinear system of viscoelastic equations

In this article, we are concerned with a system of two coupled viscoelastic equations which describes the interaction between two different fields arising in viscoelasticity. Under weaker conditions on the relaxation functions and for more general forms of nonlinearities, we extend and generalize some existing results, concerning the uniform decay for a single equation, to the case of a system.

On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

Existence results for an evolution problem with fractional nonlocal conditions

A second-order abstract problem involving derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is considered. This problem covers many situations arising in real world phenomena and extends the case of integer order to the case of non-integer order. We establish the existence of mild solutions. Under extra appropriate sufficient assumptions on the nonlinearity and on the initial data these mild solutions are shown to be classical.

How far can relaxation functions be increasing in viscoelastic problems

It is known that when we add a viscoelastic damping to a frictional damping acting in the domain we might lose the property of exponential stability of the system. Moreover, a necessary condition for a system to be sub-exponentially stable is that the kernel itself must be sub-exponentially decaying to zero. Having this in mind, a natural question to be asked is that of when this necessary condition is also sufficient. We prove that this is the case for a fairly large class of kernels.

Unboundedness for the Euler-Bernoulli beam equation with a fractional boundary dissipation

We consider the Euler-Bernoulli beam problem with some boundary controls involving a fractional derivative. The fractional derivative here represents a fractional dissipation of lower order than one. We prove that the classical energy associated to the system is unbounded in presence of a polynomial nonlinearity. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity.