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Ahmed Bchatnia

Publications and source records attributed to Ahmed Bchatnia.

8 recordsLinked to original sources
arXiv2026

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

math.AP↗
arXiv2024

Nonlinear damping effects for the 2D Mindlin-Timoshenko system

We study in this article the asymptotic behavior of the Mindlin-Timoshenko system subject to a nonlinear dissipation acting only on the equations of the rotation angles. First, we briefly recall the existence of the solution of this system. Then, we prove that the energy associated with the Mindlin-Timoshenko system fulfills a dissipation relationship showing that the energy is decreasing. Moreover, when the wave speeds are equal, we establish an explicit and general decay result for the energy.

math.AP↗
arXiv2018

Lower Bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity

In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the strong stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau--Boussouira's energy comparison principle introduced in \cite{2} (see also \cite{alabau}). One of the main advantages of these results is that they allows us to prove the optimality of the asymptotic results (as $t\rightarrow \infty$) obtained in \cite{ali}. We also extend to our model the nice results achieved in \cite{alabau} for the case of nonlinearly damped Timoshenko system with thermoelasticity. The optimality of our results is also investigated through some explicit examples of the nonlinear damping term. The proof of our results relies on the approach in \cite{AB1, AB2}.

math.AP↗
arXiv2017

Stabilization of the wave equation with moving boundary

We deal with the wave equation with assigned moving boundary ($0<x<a(t)$) upon which Dirichlet-Neuman boundary conditions are satisfied, here $a(t)$ is assumed to move slower than the light and periodically. We give a feedback which guarantees the exponential decay of the energy. The proof relies on a reduction theorem of Yoccoz. At the end we give a remark on the moving-pointwise stabilization problem.

math.DS↗
arXiv2015

Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound

In this work, we consider a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We recall first the results of well-posdness and regularity and the asymptotic behavior of the energy obtained in \cite{Ayadi}. Then, we use a fourth order finite difference scheme to compute the numerical solutions and thus we show the energy decay in several cases depending on the stability number. Résumé : Dans ce travail, on considère le système de Timoshenko non-linéaire avec Thermo-élasticité et deuxième son. On rappelle d'abord les résultats d'existence, de régularité et du comportement asymptotique de l'énergie obtenus dans \cite{Ayadi}. Ensuite, on valide numériquement ces résultats théoriques. Pour cela, on utilise une méthode de différences finies d'ordre $4$. Ainsi la solution numérique obtenue permet de valider la décroissance de l'énergie dans plusieurs cas selon la valeur du paramètre de stabilité.

math.NA↗