Papers by Abdallah el hamidi
Quasilinear PDEs, Interpolation Spaces and Hölderian mappings
Analysis Mathematica, Nov 14, 2023
Journal of Mathematical Analysis and Applications, Apr 1, 2005
We prove nonexistence results for higherorder semilinear evolution equations and inequalities of ... more We prove nonexistence results for higherorder semilinear evolution equations and inequalities of the form ∂ k u ∂t k −∆u+ λ |x| 2 u ≥ |u| q in R N ×(0, ∞), where λ ≥ − N −2 2 2. This problem can be seen as a higher-order evolution version of the nonlinear Wheeler-De Witt equation which appears in the theory of quantum cosmology. In order to show that our result is sharp in the parabolic case, we establish the existence of positive solutions to the semilinear equation ∂u ∂t − ∆u + λ |x| 2 u ≥ u q in R N × (0, ∞), for λ ≥ 0. The nonexistence results are based on the test function method, developed by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established by the construction of an explicit global solution of the semilinear parabolic inequality.
In this tutorial note, we present a spatiotemporal model for plant growth, combining two differen... more In this tutorial note, we present a spatiotemporal model for plant growth, combining two different mechanisms of competition. The first mechanism concerns the biomass growth via resources while the second concerns the spacebiomass expansion. The pure time mechanism is described by the standard underlying Kolmogorov model for interacting populations. The spatial mechanism, more adapted to plant growth, expresses the motility of each species and their capability to exclude the others from its territory.
Identification des sources dans un problème inverse d’Helmholtz par algorithme génétique
HAL (Le Centre pour la Communication Scientifique Directe), Apr 23, 2018
La méthode des nappes de sources : une résolution simplifiée de l’équation d’Helmholtz en acoustique architecturale
HAL (Le Centre pour la Communication Scientifique Directe), Apr 23, 2018
Existence of solutions for anisotropicequations with critical exponents
HAL (Le Centre pour la Communication Scientifique Directe), Jan 15, 2008
Mathematical Methods in The Applied Sciences, Feb 8, 2020
This paper introduces the use of the Proper Generalized Decomposition (PGD) method for the optica... more This paper introduces the use of the Proper Generalized Decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, i.e. optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularisation term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (i) the weak convergence based on the Brézis-Lieb decomposition, (ii) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely low image resolution and/or low Signal-To-Noise Ratio (SNR).
arXiv (Cornell University), Jul 5, 2019
The interpolation on Grassmann manifolds in the framework of parametric evolution partial differe... more The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.
Differential Equations and Applications, 2022
In this paper, we study the existence of positive solutions for the Kirchhoff equations with conc... more In this paper, we study the existence of positive solutions for the Kirchhoff equations with concave terms ⎧ ⎪ ⎨ ⎪ ⎩ − a + b Ω |∇u| 2 dx Δu = f (x,u) − λ |u| q−2 u, in Ω, u = 0, on ∂ Ω, (0.1) where Ω is a bounded domain with a C 2-boundary ∂ Ω in R N (N = 1,2,3) , and a,b > 0 , 1 < q < 2. By applying variational methods, we show that there exists a constant λ * > 0 such that for any λ ∈ (0,λ *) , problem (0.1) has at least two positive solutions.
Existence and nonexistence results for reaction-diffusion equations in product of cones
Central European Journal of Mathematics, Mar 1, 2003
... u(x; 0) = u0(x) ¶ 0 in RN ¤ E-mail: [email protected] y E-mail: [email protected] Page 2... more ... u(x; 0) = u0(x) ¶ 0 in RN ¤ E-mail: [email protected] y E-mail: [email protected] Page 2. 62 A. El Hamidi, GG Laptev / Central European Journal of Mathematics 1 (2003) 61{78 ... Page 3. A. El Hamidi, GG Laptev / Central European Journal of Mathematics 1 (2003) 61{78 63 ...
Nonexistence of solutions of semilinear inequalities in cone-like domains
Bulletin of the Belgian Mathematical Society, Simon Stevin
Theoretical Computer Science, Nov 1, 1997
We consider a three-dimensional thermal-diffusion model for a premixed burner flame. Many experim... more We consider a three-dimensional thermal-diffusion model for a premixed burner flame. Many experimental and theoretical works in condensed-phase and gas combustion show that the flame front may propagate in a number of different ways. The structure and stability properties of the front depend essentially on the physical parameters of the model. This article describes the use of the symbolic manipulation language MAPLE for the analysis of bifurcation phenomena in gas combustion. It shows how symbolic manipulation languages can be combined effectively with analysis and numerical computations for this type of investigation.
Ecological Modelling, Jun 1, 2012
Ricerche Di Matematica, Jul 1, 2006
We study a perturbed anisotropic equation without using the knowledge of the limiting problem. Th... more We study a perturbed anisotropic equation without using the knowledge of the limiting problem. This provides a different method from that introduced by Brzis and Nirenberg [4]. Our arguments use some tools recently developed in [5, 6].
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Oct 1, 2007
The existence of multiple nonnegative solutions to the anisotropic critical problem − N i=1 ∂ ∂x ... more The existence of multiple nonnegative solutions to the anisotropic critical problem − N i=1 ∂ ∂x i ∂u ∂x i p i −2 ∂u ∂x i = |u| p * −2 u in R N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentrationcompactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions S is included in L ∞ (R N) and is located outside of a ball of radius τ > 0 in L p * (R N). Résumé Nous montrons l'existence d'une infinité de solutions positives pour le problème anisotropique avec exposant critique. La méthode consistè a regarder la meilleure constante d'une inégalité du type Poincaré-Sobolev età adapter le fameux principe de concentration-compacité de P.L. Lions. De plus, on montre que l'ensemble des solutions S est contenu dans L ∞ (R N) et est localisé en dehors d'une boule de rayon τ > 0 dans L p * (R N).
Differential and Integral Equations, 2007
One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is... more One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work [7], Brézis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact.
Differential and Integral Equations, 2005
A compactness result is revised in order to prove the pointwise convergence of the gradients of a... more A compactness result is revised in order to prove the pointwise convergence of the gradients of a sequence of solutions to a general quasilinear inequality (anisotropic or not, degenerate or not) and for an arbitrary open set. Combining this result with the well-known Brézis-Lieb lemma, we derive simple proofs of Palais-Smale properties in many optimization problems especially on unbounded domains.
Mathematics, Jul 25, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Ecological Modelling, Jun 1, 2012
We present a spatial competition model for clonal plant growth that combines two different mechan... more We present a spatial competition model for clonal plant growth that combines two different mechanisms of competition. The first one is described by the standard underlying Kolmogorov Model for two interacting populations. A second competition mechanism, more specific to clonal plant growth, expresses the motility of each species and their capacity to resist to competitor's space intrusion. This model leads to a degenerated nonlinear reaction-diffusion system in which the diffusion coefficient of each species vanishes wherever the other species is beyond a certain biomass value. Indeed, we show that pattern forming instability can not occur in general competition systems if the exclusion ability of both species are small even if the diffusions are degenerated. The effect of diffusion degeneracy on patterns formation is carried out. Numerical simulations of this two level competition model are performed when the reaction terms are given by the competitive Lotka-Volterra equations. We finally discuss the potential of such nonlinear reaction-diffusion systems to be a surrogate model for phalanx-guerilla competition.
International Journal for Numerical Methods in Fluids, Apr 5, 2021
The interpolation on Grassmann manifolds in the framework of parametric evolution partial differe... more The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.
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Papers by Abdallah el hamidi