Mustapha Jazar

Lebanese University · Laboratoire de Mathématiques et Applications

In this paper, we consider the Föppl‐von Kàrmàn equations in the case of a simply supported thin plate. We introduce a nonlinear Gauss‐Seidel fixed point scheme which allows to obtain a constructive proof of the existence and the uniqueness, when a small nonzero source term is considered. Numerical simulations are given using finite elements approx...

This paper deals with an inverse source problem for an elliptic equation, using interior measurements. Its motivation lies in the seawater intrusion phenomenon, where we are interested in identifying point sources representing illegal wells. A cost function transforming our inverse problem into an optimization one is proposed, and numerical results...

This article is devoted to prove the existence of a non-negative solution for a degenerate parabolic strongly coupled system which arises from seawater intrusion model in confined aquifers. An approximate linear discretized in time scheme is then set up yielding to the existence result with no restriction on the space dimension. The availability of...

As a major hotspot of climate change, Lebanon suffers from a water resources crisis enhanced by the increase of anthropogenic activities. In this paper, the impacts of climate change and of the Syrian refugee crisis are combined with the impact of demographic growth to assess their aggregated impact on seawater intrusion in the Tripoli aquifer. A h...

The approximation of solutions to partial differential equations by tensorial separated representations is one of the most efficient numerical treatment of high dimensional problems. The key step of such methods is the computation of an optimal low-rank tensor to enrich the obtained iterative tensorial approximation. In variational problems, this s...

This paper deals with an inverse monopolar source problem for the Poisson equation, from interior measurements, whose motivation lies in the sea water intrusion phenomenon. Global logarithmic stability estimates of locations and intensities for monopolar sources in this equation are established. To do that, we make an appropriate choice of a test f...

Water resources in Mediterranean coastal aquifers are subject to overexploitation leading to an increase in seawater intrusion. Based on the United Nations Environment Program, "UNEP" 75% of people in the world will live in coastal cities by 2020. This is having a major impact on the salinization process. This paper deals with the impact of demogra...

This paper deals with an inverse monopolar source problem for the screened Poisson equation (Formula presented.), (Formula presented.), from interior measurements, whose motivation lies in the seawater intrusion phenomenon. New Lipschitz stability estimates of locations and intensities for monopolar sources, with explicit stability constants, are e...

In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier-Stokes equations for a 2D incompressible fluid flow. We present a nu...

From mathematical and numerical points of view, we study the sensitivity with respect to parametric evolutions, of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition (POD) basis determined once and for all, with association to a fixed parameter. More precisely, we will be con...

In this paper, we describe a statistical shape analysis founded on a robust elastic metric. The proposed metric is based on geodesics in the shape space. Using this distance, we formulate a variational setting to estimate intrinsic mean shape viewed as the perfect pattern to represent a set of given shapes. By applying a geodesic-based shape warpin...

We prove existence and uniqueness of radial weak solutions of the doubly nonlinear parabolic equation ut=div[D(u)ϕ(∇u)],ut=div[D(u)ϕ(∇u)], D ϕϕ D(s)=s(1-s)D(s)=s(1-s) ϕ(p)≔p/(1+|p|2)ϕ(p)≔p/(1+|p|2)

In this paper, we study degenerate parabolic systems, which are strongly coupled. We prove general existence results, but the uniqueness remains an open question. Our proof of existence is based on a crucial entropy estimate which both control the gradient of the solution and the non-negativity of the solution. Our systems are of porous medium type...

In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. A POD basis associated with a set of reference values of the characteristic parameters is considered. From this basis, a parametric reduced order model (ROM) projecting the initial equati...

In this paper, we consider the flow of fresh- and saltwater in a saturated porous medium in order to describe the process of seawater intrusion. Starting from a formulation with constant densities of fresh-and saltwater, whose velocities are proportional to the gradient of pressure (Darcy's law), we consider the formal asymptotic limit as the aspec...

In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation.

We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_{t}=F(D^2u,D u,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applicatio...

We are interested in the mathematical study of the sensitivity of a reduced order model (ROM) of a particular single-parameterised quasi-linear equation, via the parametric evolution. More precisely, the ROM of interest is obtained in two different ways: First, we reduce the complete parametric equation using a proper orthogonal decomposition (POD)...

In this article, we consider a simple one-dimensional variational model, describing the delamination of thin films under cooling. We characterize the global minimizers, which correspond to films of three possible types: non delaminated, partially delaminated (called blisters), or fully delaminated. Two parameters play an important role: the length...

In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. We show the existence of a bounded positive invariant and attracting set. The possibility of existence and uniqueness of positive equilibrium are considered. The asymptotic behavior of the positive equilibrium and the existence of Hopf-bifu...

In this paper, we study the standard one-dimensional (non-overdamped) Frenkel--Kontorova (FK) model describing the motion of atoms in a lattice. For this model we show that for any supersonic velocity $c>1$, there exist bounded traveling waves moving with velocity $c$. The profile of these traveling waves is a phase transition between limit states...

We study Föppl-von Kármán system (1) that modelize delamination of embedded compressed thin film. We prove that if the radial component of the stress tensor is less then or equal the pre-stress, then system (1) has no nontrivial regular radial solution under suitable boundary clamped conditions.

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.

In this paper, we discuss a numerical approximation of the first eigenvalue of the p-Laplace operator on the sphere (S n, g) of R n+1.

We study the evolution equation ∂ t h=-ν∂ x 2 h-K∂ x 4 h+λ 1 (∂ x h) 2 -λ 2 ∂ x 2 (∂ x h) 2 · which was introduced by J. Muñoz-Garcia, R. Cuerno and M. Castro [“Short-range stationary patterns and long-range disorder in an evolution equation for one-dimensional interfaces”, Phys. Rev. E 74, No. 5, 050103(R) (2006), doi:10.1103/PhysRevE.74.050103] i...

We study local existence and uniqueness in the phase space Hμ×Hμ−1(RN) of the solution of the semilinear wave equation utt−Δu=ut|ut|p−1 for p>1.

It is well known that minimization problems involving sublinear regularization terms are ill-posed, in Sobolev spaces. Extended results to spaces of bounded variation functions BV were recently showed in the special case of bounded regularization terms. In this note, a generalization to sublinear regularization is presented in BV spaces. Notice tha...

In this paper, we consider the flow of fresh and saltwater in a saturated porous medium in order to describe the seawater intrusion. Starting from a formulation with constant densities respectively of fresh and of saltwater, whose velocity is proportional to the gradient of pressure (Darcy's law), we consider the formal asymptotic shallow water lim...

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set sigma(A) = {i lambda(k); k is an element of Z*} is discrete and satisfies Sigma 1/vertical bar lambda(k)vertical bar(l)delta(n)(k) < infinity, where l is a nonnegative integer and delta(k) = min(vertical bar lambda(k+1)-l...

We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equa...

We study a strongly coupled system consisting of a parabolic equation and a singular Hamilton–Jacobi equation in one space dimension. This system describes the dynamics of dislocation densities in a material submitted to an exterior applied stress. Our system is a natural extension of that studied in [1616. Ibrahim , H. ( 2009 ). Existence and uniq...

We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\mathbb{R}^N))$ of the semilinear wave equation $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.

In this paper, the authors considered the equation: The proof for which relied on the existence of a Lyapunov functional w in self-similar variables y and s. There was a mistake in the equation satisfied by w (y, s): which caused the results of the paper to be incorrect. The paper has therefore been retracted by the authors.

We study the existence of (distribution/viscosity) solutions of a singular parabolic/Hamilton-Jacobi coupled system. Our motivation stems from the study of the dynamics of dislocation densities in a crystal of finite size. The method of the proof consists in considering a parabolic regularization of the system, and then passing to the limit after o...

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equation...

We study the existence and the properties of reduced measures for the parabolic equations ∂ t u − Δu + g(u) = 0 in Ω × (0, ∞) subject to the conditions (P): u = 0 on ∂Ω × (0, ∞), u(x, 0) = µ and (P′): u = µ′ on ∂Ω × (0, ∞), u(x, 0) = 0, where µ and µ′ are positive Radon measures and g is a continuous nondecreasing function.

In this paper we give a positive answer to the conjecture proposed in [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (1) (2007) 17–39] by El Soufi et al. concerning the finite time blow-up for...

We study the existence of singular solutions to the equation −div(|Du|p−2Du)=|u|q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.

In this paper we consider the semi-linear wave equation: $u_{tt}-\Delta u=u_t|u_t|^{p-1}$ in $\mathbb{R}^N$. We provide an associated energy. With this energy we give the blow-up rate for blowing up solutions in the case of bounded below energy.

In this paper we consider the semi-linear wave equation: utt − Δu = ut|ut|p−1 in where and p < 3 if N = 1, p ≠ 3 if N = 2. We give an energetic criteria and optimal lower bound for blow-up solutions of this equation.

We give sufficient conditions on the initial data so that a semilinear wave inequality blows up in finite time. Our method is based on the study of an associated second-order differential inequality. The same method is applied to some semilinear systems of mixed type.

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L∞-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship be...

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\frac\pi 2$, $0\le v <\pi$, is the \emph{unique} extremal metric of the first eigenvalue of t...

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of...

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if $A$ generates a polynomially bounded $n$-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_k; k\in\mathbb{Z}^* \}$ is discrete and satisfies $\sum \frac{1}{|\lambda_k|^\ell\delta_k^n}<\infty$ ($n$ and $\ell$ nonne...

The aim of this paper is to study numerically an alternative method of the fictitious domains technique. We show the convergence and the consistency of the new method.

This paper highlights a new short proof of a generalization of Gelfand's theorem through the use of spectral distributions. But above all it aims at studying, when the spectrum is discrete, if the operator admits a spectral resolution of the identity.

In this chapter we study some problems of Spectral Theory for pseudo-differential operators with hypoelliptic symbols in the classes S(M; Ф, Ψ) considered in Chapter 1; see in particular Sections 1.1 and 1.3.1.

In this work we give a new criteria for the existence of periodic and almost periodic solutions for some differential equation in a Banach space. The linear part is nondensely defined and satisfies the Hille-Yosida condi-tion. We prove the existence of periodic and almost periodic solutions with condition that is more general than the known exponen...

We study the existence of singular separable solutions to the 2-dimensional quasilinear equation −∇·(|∇u|p−2∇u)+|u|q−1u=0 under the form u(r,θ)=r−βω(θ). We obtain the full description of the set of such solutions by combining a 2-dimensional shooting method with a phase plane analysis approach.

Consider the equation u '' +|u| p-1 u=b|u ' | q-1 u ' ,t≥0, where p,q>1 and b>0 are real numbers. We investigate the critical case q=2p/(p+1). We prove that all nontrivial solutions blow-up in finite time and that the asymptotic behavior near blow-up exhibits a strong dependence upon the values of b. Namely, (a) if b≥b 1 (p):=(p+1)((p+1)/2p) (p+1)...

In this paper we give a simple representation of the solution of the Cauchy problem when the operator admits a spectral distribution. First we apply this to the Schrödinger operator on RN with and without potential, and then on a bounded domain. In this case we give the expression of the associated spectral distribution. A second application is tha...

We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of A n , for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the...

We introduce a C-regularized scalar operator. These have the properties of spectral operators of scalar type, except that the spectral measure is bounded and countably additive only after applying the regularizing operator C. We discuss the relationship between regularized scalar operators, regularized functional calculi, and generating a polynomia...

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distri...

We construct fractional and imaginary powers for the positive momentum B of a spectral distribution and prove the basic properties. The main result is that for any α>0, -B α generates a bounded strongly continuous holomorphic semigroup of angle π 2. In particular for α=1, using Stone’s generalized theorem, if iB generates a k-times integrated group...

In this paper we construct fractional and imaginary powers for the positive momentum B of a spectral distribution and prove the basic properties. The main result is that for any generates a bounded strongly continuous holomorphic semigroup of angle. In particular for a = 1, using Stone's generalized theorem, if iB generates a k-times integrated gro...

In this paper, we introduce the notion ofspectral distribution which is a generalization of the spectral measure. This notion is closely related to distribution semigroups and generalized scalar operators. The associated operator (called themomentum of the spectral distribution) has a functional calculus defined for infinitely differentiable functi...