arXiv subjects

Hatem Zaag

Publications and source records attributed to Hatem Zaag.

At least 19 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 $$ \partial_t^2 u - \partial_x^2 u + \frac{\mu}{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ 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 ($\mu =0$).

math.AP↗

arXiv2025

Feedback approximate controllability of blowup points for the heat equation with anti-interference blowup profile

This paper is concerned with a feedback approximate controllability problem of blowup points for the heat equation. We show that the system is approximately controllable for blowup points with feedback controls and the feedback operator is bounded at any time before blowup. It is also proved that the blowup profile for feedback controllability of blowup points is stable with respect to initial data. That is, suppose that the initial data has a very small perturbation, the blowup profiles also have tiny changes. More precisely, it just undergoes a tiny translation in space and time. This means that our feedback strategy is anti-interference.

math.OC↗

arXiv2025

Scalar behavior for a complex multi-soliton arising in blow-up for a semilinear wave equation

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution behaves exactly as in the real-valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex-valued first order Toda system governing the evolution of the positions and the phases of the solitons.

math.AP↗

arXiv2024

Radial blow-up standing solutions for the semilinear wave equation

We consider the semilinear wave equation with a power nonlinearity in the radial case. Given $r_0>0$, we construct a blow-up solution such that the solution near $(r_0,T(r_0))$ converges exponentially to a soliton. Moreover, we show that $r_0$ is a non-characteristic point. For that, we translate the question in self-similar variables and use a modulation technique. We will also use energy estimates from the one dimensional case treated by Merle and Zaag in 2007. Of course because of the radial setting, we have an additional gradient term which is delicate to handle. That's precisely the purpose of our paper.

math.AP↗

arXiv2024

Standing Sphere Blow-up Solutions to The Nonlinear Heat Equation

In this paper, we construct a singular standing ring solution of the nonlinear heat in the radial case. We give rigorous proof for the existence of a ring blow-up solution in finite time. This result was predicted formally by Baruch, Fibich and Gavish \cite{BFGpd10}. We also prove the stability of these dynamics among radially symmetric solutions.

math.AP↗

arXiv2024

A better bound on blow-up rate for the superconformal semilinear wave equation

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $|\log(T-t)|^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.

math.AP↗

arXiv2023

Rescaling method for blow-up solutions of nonlinear wave equations

We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A careful study of the convergence of the proposed scheme is carried out and several numerical examples are performed in illustration.

math.NA↗

arXiv2023

The blow-up rate for a loglog non-scaling invariant semilinear wave equation

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.

math.AP↗

arXiv2023

Flat blow-up solutions for the complex Ginzburg Landau equation

In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for $\beta$ and $\delta$. Specifically, for a fixed $\beta \in \mathbb{R}$, the existence of finite-time blow-up solutions for arbitrarily large values of $ |\delta| $ is still unknown. According to a conjecture made by Popp et al. \cite{POPphd98}, when $\beta = 0$ and $\delta$ is large, blow-up does not occur for \textit{generic initial data}. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for $\beta = 0$ and any $\delta \in \mathbb{R}$ with different types of blowup.

math.AP↗

arXiv2023

Instabilities Appearing in Cosmological Effective Field theories: When and How?

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = α(\partial_x u(x,t))^2 +β\partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $α>0$. We study the nature of this divergence as a function of the parameters $α>0 $ and $β\ge0$. The divergence does not disappear even when $β$ is very large contrary to what one might believe (note that since we consider fixed initial data, $α$ and $β$ cannot be scaled away). But it will take longer to appear as $β$ increases when $α$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.

math-ph↗

arXiv2023

Unsupervised physics-informed neural network in reaction-diffusion biology models (Ulcerative colitis and Crohn's disease cases) A preliminary study

We propose to explore the potential of physics-informed neural networks (PINNs) in solving a class of partial differential equations (PDEs) used to model the propagation of chronic inflammatory bowel diseases, such as Crohn's disease and ulcerative colitis. An unsupervised approach was privileged during the deep neural network training. Given the complexity of the underlying biological system, characterized by intricate feedback loops and limited availability of high-quality data, the aim of this study is to explore the potential of PINNs in solving PDEs. In addition to providing this exploratory assessment, we also aim to emphasize the principles of reproducibility and transparency in our approach, with a specific focus on ensuring the robustness and generalizability through the use of artificial intelligence. We will quantify the relevance of the PINN method with several linear and non-linear PDEs in relation to biology. However, it is important to note that the final solution is dependent on the initial conditions, chosen boundary conditions, and neural network architectures.

cs.LG↗

arXiv2022

Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation $$\frac{\partial h}{\partial t}=\frac{\partial^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^β},\quadβ>0$$ subject to initial boundary value conditions $$h(\cdot,0)=h_0>0,\quad h(\pm1,t)=1.$$ We derive the intermediate extinction profile with refined asymptotics, and with extinction time $T$ and extinction point $0$, the gradient profile behaves as $x\rightarrow0$ like $$\lim_{t\rightarrow T}\,(\nabla h)(x,t)\quad\sim\quad\frac{1}{\sqrt{2β}}\frac{x}{|x|}\frac{1}{\sqrt{|\log|x||}}\left[\frac{(β+1)^2}{8β}\frac{|x|^2}{|\log|x||}\right]^{\frac{1}{β+1}-\frac12},$$ agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.

math.AP↗

arXiv2022

Modulation theory for the flat blowup solutions of nonlinear heat equation

In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a at blowup profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well celebrated method, based on the reduction of the problem to a finite dimensional one, then the use of a topological shooting method to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.

math.AP↗

arXiv2022

On degenerate blow-up profiles for the subcritical semilinear heat equation

We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.

math.AP↗

arXiv2022

The blow-up rate for a non-scaling invariant semilinear heat equation

We consider the semilinear heat equation $$\partial_t u -Δu =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

math.AP↗

arXiv2021

Refined blow-up asymptotics for a perturbed nonlinear heat equation with a gradient and a non-local term

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1, 2], we constructed a solution $u$ for that equation such that $u$ and $\nabla u$ both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.

math.AP↗

arXiv2021

Classification of the blow-up behavior for a semilinear wave equation with nonconstant degenerate coefficients

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.

math.AP↗

arXiv2021

Feedback controllability for blowup points of heat equation

This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $Ω$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $ω$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $a\in ω$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any feedback control, $a\in Ω\setminus \overlineω$ could not be the unique blowup point.

math.OC↗