arXiv subjects

Tahir Boudjeriou

Publications and source records attributed to Tahir Boudjeriou.

7 recordsLinked to original sources

arXiv2026

Twisted Gradient Flow Approach to Vegetation-Rainfall-Bushfire Interactions Model

We consider a vegetation-rainfall-bushfire interaction model consisting of two partial differential equations describing the evolutions of bushfire intensity and water availability, together with an ordinary differential equation governing the vegetation density. The main feature of the model is that the ODE contains a nonlocal coefficient multiplying the vegetation density, which makes the mathematical analysis highly nontrivial. In this paper, we regard this nonlocal coefficient as the generator of a deformation of the metric structure of the underlying real Hilbert space. This interpretation enables us to formulate the ODE as a twisted gradient flow on state-dependent Hilbert spaces and to establish the existence of strong solutions to the associated initial-boundary value problem.

math.AP↗

arXiv2026

Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering $W^{1,q}$-small initial data

In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_{0}, u_{0t})=(u, u_{t})|_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_{t})$ in $ W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.

math.AP↗

arXiv2025

Asymptotic issue for fractional laplacian on long cylinders

In this paper, we are concerned with the asymptotic behavior of weak solutions to certain elliptic and parabolic problems involving the fractional $p$-Laplacian in cylindrical domains that become unbounded in one direction. The nonlocal nature of the operator describing the equations creates several technical difficulties in treating problems of this type. The main results, obtained within a nonlocal abstract framework, extend and complement related properties established in the local setting.

math.AP↗

arXiv2021

Some qualitative properties for the Kirchhoff total variation flow

In this paper we are concerned with the following Kirchhoff type problem involving the 1-Laplace operator : \begin{equation*} \left\{\begin{array}{llc} u_{t}-m\left(\int_Ω|Du|\right)Δ_{1} u=0 & \text{in}\ & Ω\times (0,+\infty) , \\ u=0 & \text{on} &\partial Ω\times (0,+\infty),\\ u(x,0)=u_{0}(x) & \text{in} &Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^{N}$ ($N\geq 1$) is a bounded smooth domain, $m :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is an increasing continuous function that satisfies some conditions which will be mentioned further down, and $Δ_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$ denotes the 1-Laplace operator. The main purpose of this work is to investigate from the initial data $u_{0}$ and the nonlinear function $m$ the existence and asymptotic behavior of solutions near the extinction time.

math.AP↗

arXiv2020

Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian

In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.

math.AP↗

arXiv2020

Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional $p$-Laplacian with logarithmic nonlinearity \begin{equation*}\label{eq}\left\{ \begin{array}{llc} u_{t}+(-Δ)^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log(|u|) & \text{in}\ & Ω,\;t>0 , u =0 & \text{in} & \mathbb{R}^{N}\backslash Ω,\;t > 0, u(x,0)=u_{0}(x), & \text{in} &Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N \, ( N\geq 1)$ is a bounded domain with Lipschitz boundary and $2\leq p< \infty$. The local existence will be done by using the Galerkin approximations. By combining the potential well theory with the Nehari manifold we establish the existence of global solutions. Then, by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional $p$-Laplacian.

math.AP↗

arXiv2020

Existence of solution for a class of nonlocal problem via dynamical methods

In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_Ωg(u)\,dx \right)Δu =f(u), \quad x \in Ω\\ u=0, \hspace{2 cm} x \in \partial Ω, \end{array}\right. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N \, ( N \geq 2)$ is a smooth bounded domain and $a:\overlineΩ \times \mathbb{R} \to \mathbb{R}$ and $g,f: \mathbb{R} \to \mathbb{R}$ are $C^1$-functions that satisfy some technical conditions.

math.AP↗