Mokhtar Kirane

Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

Abstract We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.

On oscillation of nonlinear second order differential equation with damping term

We present new oscillation criteria for the second order nonlinear differential equation with damping. Our theorems are stated in general form; they complement and extend related results known in the literature. The relevance of our results is illustrated with a number of examples. To facilitate computations in these examples, we use Mathematica^(R), a symbolic computer language.

Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation

In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we valid the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.

An inverse problem for a generalized fractional diffusion

We propose a method for determining the solution and source term of a generalized time-fractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L 2 space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag-Leffler function is used to relax the smoothness requirement on these conditions.

Non-Existence Results for a Semilinear Hyperbolic Problem with Boundary Condition of Memory Type

We consider a problem which models the evolution of sound in a compressible fluid with reflection of sound at the surface of the material. Different methods such as the concavity method of Levine, the potential well method and an argument due to Tsutsumi are used to derive global non-existence theorems.

Lyapunov-type inequalities for fractional partial differential equations

Abstract We present Lyapunov inequalities for the fractional p-Laplacian ( − Δ p ) s , 1 p ∞ , s ∈ ( 0 , 1 ) , in an open bounded set Ω ⊂ R N , under homogeneous Dirichlet boundary conditions. We discuss two cases, the case s p > N and the case s p N . We use the obtained inequalities to provide lower bounds for the first eigenvalue of the fractional p-Laplacian. To the best of our knowledge, this is the first work dealing with Lyapunov-type inequalities for fractional partial differential equations.

Exponential Growth for a Fractionally Damped Wave Equation

We consider a nonlinear wave equation with an internal damping represented by a fractional time derivative and with a polynomial source. It is proved that the solution is unbounded and grows up exponentially in the Lp-norm for sufficiently large initial data. To this end we use some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood inequality.

Fujita’s Exponent for a Semilinear Wave Equation with Linear Damping

Abstract We discuss the critical exponent problem for the semilinear wave equation with linear damping utt +(-1)m|x|αΔmu + ut = f(t,x)|u|p + w(t,x), t > 0, x ∈ℝn,and show that it coincides with the Fujita exponent for the heat equation with the same nonlinearity.

Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping

utt −∆u+D α +u = h(t, x) |u| p posed in Q := (0,∞) × RN , where Dα +u, 0 < α < 1 is a time fractional derivative, with given initial position and velocity u(0, x) = u0(x) and ut(0, x) = u1(x). We find the Fujita’s exponent which separates in terms of p, α and N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u1(x) and h(x, t) assuring non-existence of local solutions.

Control and optimal response problems for quasilinear impulsive integrodifferential equations

Abstract In various real-world applications, there is a necessity given to steer processes in time. More and more it becomes acknowledged in science and engineering, that these processes exhibit discontinuities. Our paper on theory of control (especially, optimal control) and on theory of dynamical systems gives a contribution to this natural or technical fact. One of the central results of our paper is the Pontryagin maximum principle [L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, John Wiley, New York, 1962] which is considered in sufficient form for the linear case of impulsive differential equations. The problem of controllability of boundary-value problems for quasilinear impulsive system of integrodifferential equations is investigated. The control consists of a piecewise continuous function part as well as impulses which act at a variable time. By studying the optimal control of response, we give a first inclusion of an objective function. By this pioneering contribution, we invite to future research in the wide field of optimal control with impulses and in modern challenging applications.