Browsing by Author "Boudjeriou, Tahir"

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Asymptotic behavior of parabolic nonlocal equations in cylinders becoming unbounded
(Springer, 2023) Boudjeriou, Tahir
The goal of this paper is to discuss the asymptotic behavior of weak solutions to a class of parabolic equations involving fractional Laplacian in cylindrical domains becoming unbounded in one direction. The results presented in this paper are new and extend some main results in the literature for local and nonlocal elliptic problems with Dirichlet boundary condition
Asymptotic Issues for Fractional Laplacian on Long Cylinders
(Birkhauser, 2025) Boudjeriou, Tahir; Roy, Prosenjit
n 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
Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity
(Wiley, 2024) Boudjeriou, Tahir
The main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as t$t$ goes to infinity for the following class of parabolic Kirchhoff equations: ut+∥u∥(θ−1)Ns(−Δ)N/ssu=λ|u|q−2uexpα0|u|NN−s|x|γinΩ,t>0,u=0inRN∖Ω,t>0,u(x,0)=u0(x)inΩ,$$\begin{eqnarray*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{llc}u_{t}+\Vert u\Vert ^{\frac{(\theta -1)N}{s}}(-\Delta)^{s}_{N/s}u=\frac{\lambda |u|^{q-2}u\exp {\left(\alpha _{0}|u|^{\frac{N}{N-s}}\right)}}{|x|^{\gamma }} &\text{in}\ &\Omega,\;t>0, \\ u =0 &\text{in} & \mathbb {R}^{N}\backslash \Omega,\;t > 0, \\ u(x,0)=u_{0}(x)& \text{in} &\Omega, \end{array} \right.} \end{eqnarray*}$$where ∥u∥Ns=∫R2N|u(x,t)−u(y,t)|N/s|x−y|2Ndxdy,$$\begin{equation*} \hspace*{7pc}\Vert u\Vert ^{\frac{N}{s}}=\int _{\mathbb {R}^{2N}}\frac{|u(x,t)-u(y,t)|^{N/s}}{|x-y|^{2N}}\,dxdy, \end{equation*}$$Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \, (N\ge 2)$ is a bounded domain with a Lipschitz boundary, 0∈Ω$0\in \Omega$, α0,λ>0$\alpha _{0},\lambda >0$, θ≥1$\theta \ge 1$, γ∈[0,N)$\gamma \in [0, N)$, q>Nθ/s$q>N\theta /s$, and (−Δ)N/ss$(-\Delta)_{N/s}^{s}$ is the fractional N/s$N/s$‐Laplacian operator, s∈(0,1)$s\in (0,1)$.
Asymptotics for a wave equation with critical exponential nonlinearity
(Elsevier Ltd, 2024) Boudjeriou, Tahir; Van Thin, Nguyen
In this paper, we discuss some qualitative analysis of solutions to the following Cauchy problem of wave equations involving the 1/2-Laplace operator with critical exponential nonlinearity [Formula presented] where λ>0, δ≥0, q>2, and α0>0. By using the contraction mapping principle, we show that the above Cauchy problem has a unique local solution. With the help of the potential well argument, we characterize the stable sets by the asymptotic behavior of solutions as t goes to infinity, as well as the unstable sets by the blow-up of solutions in finite time.
Decay estimates and extinction properties for some parabolic equations with fractional time derivatives
(Springer Nature, 2024) Boudjeriou, Tahir
The main goal of this paper is to study the asymptotic behaviour and the finite extinction time of weak solutions to some time-fractional parabolic equations. Moreover, we improve some results in [5, 10] by dropping out some conditions assumed there.
Existence and uniqueness of solution for some time fractional parabolic equations involving the 1-Laplace operator
(Springer, 2023) Alves, Claudianor O.; Boudjeriou, Tahir
This paper is devoted to study the time fractional parabolic 1-Laplacian type. Firstly, by using the parabolic regularization technique and approximating the parabolic 1-Laplacian problem by a class of parabolic equations of p-Laplacian type with p> 1 , we establish the existence of global weak radial solutions to the considered problem for a wide class of nonlinearities. Secondly, we discuss the extinction property of solutions to the time fractional total variation flow (FTVF) with different boundary conditions (Dirichlet and Neumann conditions). We conclude this paper by providing an example of explicit solution to the (FTVF)
Existence of solution and qualitative behavior for a class of heat equations
(Elsevier, 2024) Alves, Claudianor O.; Boudjeriou, Tahir; Prado, Humberto
The objective of this paper is to investigate the existence of solutions and their qualitative behavior for a given class of nonlinear evolutionary equations. We demonstrate that the pseudo-differential operator Δexp⁡(−cΔ) acts as the infinitesimal generator of the solution operator. Here, Δ denotes the Euclidean Laplace operator, and c is a positive constant. We establish the appropriate domain for the operator Δexp⁡(−cΔ) and prove that it generates a C0 semigroup on L2(RN). Additionally, we introduce a scale of spaces wherein smooth solutions exist, and these spaces are continuously embedded into the Sobolev class. Finally, we investigate the nonlinear evolution problem for a broad class of nonlinearities.
Existence of solution for a class of heat equation involving the 1-Laplacian operator
(Elsevier, 2022) Alves, Claudianor O.; Boudjeriou, Tahir
This paper concerns the existence of global solutions for the following class of heat equations involving the 1-Laplacian operator for the Dirichlet problem {ut−Δ1u=f(u)inΩ×(0,+∞),u=0in∂Ω×(0,+∞),u(x,0)=u0(x)inΩ, where Ω⊂RN is a smooth bounded domain, N≥1, f:R→R is a continuous function satisfying some technical conditions and [Formula presented] denotes the 1-Laplacian operator. The existence of global solutions is done by using an approximation technique that consists in working with a class of p-Laplacian problems associated with (P) and then taking the limit when p→1+ to get our results
Global existence and some qualitative properties of weak solutions for a class of pseudo-parabolic equations with a logarithmic nonlinearity in whole RN
(Cambridge University Press, 2025) Alves, Claudianor; Boudjeriou, Tahir
In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as t→+∞ . Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay
Global existence for parabolic p-Laplace equations with supercritical growth in whole RN
(Elsevier, 2024) Alves, Claudianor O.; Boudjeriou, Tahir
The focus of this paper is to investigate the global existence and uniqueness of weak solutions for a class of parabolic p-Laplacian equations whose nonlinearity has a supercritical growth.
A note on a supercritical p-Laplacian equation with logarithmic perturbation
(Elsevier, 2024) Boudjeriou, Tahir
In this short note, we study the existence of ground state solutions for p-Laplacian elliptic problems with supercritical growth and logarithmic nonlinearity, both within the interior and on the boundary of a domain.
Some qualitative properties for the Kirchhoff total variation flow
(Taylor and Francis Group, 2022) Boudjeriou, Tahir
The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem