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Boumediene Abdellaoui

Publications and source records attributed to Boumediene Abdellaoui.

At least 19 recordsLinked to original sources
arXiv2025

Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation $ w_t+(-Δ)^sw= h\;;\; w(x,t)=0 \text{ in } \; (\mathbb{R}^N\setminusΩ)\times(0,T)\;;\; w(x,0)=w_0(x) \; \text{in}\; Ω$, where $Ω$ is an open bounded subset of $\mathbb{R}^N$. The proof is based on a new pointwise estimate on the fractional gradient of the corresponding kernel. Moreover, we establish the compactness of $(w_0,h)\mapsto w$. As a majeur application, in the second part , we establish existence and regularity of solutions to a class of Kardar--Parisi--Zhang equations with fractional diffusion and a nonlocal gradient term. Additionally, several auxiliary results of independent interest are obtained.

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arXiv2024

Fractional heat equation involving Hardy-Leray Potential

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-Δw)^s=\fracλ{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases} $$ where $N> 2s$, $0 p_+(λ,s)$. Then there are not any non-negative supersolutions. - Let $p<p_+(λ,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let $ 1< p\le F(λ,s)$. Here we show that a weighted norm of any positive solution blows up in finite time. - Let $F(λ,s)<p<p_+(λ,s)$. Here we prove the existence of global solutions under suitable hypotheses.

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arXiv2023

Global fractional Calderón-Zygmund type regularity

We obtain a global fractional Calderón-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $Ω\subset \mathbb{R}^N$, $N \geq 2$, a bounded domain with boundary $\partial Ω$ of class $C^2$, $s \in (0,1)$ and $f \in L^m(Ω)$ for some $m \geq 1$, we consider the problem $$ \left. \begin{aligned} (-Δ)^s u = f \quad \mbox{in } Ω, \qquad\ u = 0 \quad \mbox{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. $$ and, according to $m$, we find the values of $s \leq t < \min\{1,2s\}$ and of $1 < p < +\infty$ such that $u \in L^{t,p}(\mathbb{R}^N)$ and such that $u \in W^{t,p}(\mathbb{R}^N)$.

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arXiv2022

Deterministic KPZ-type equations with nonlocal "gradient terms"

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar-Parisi-Zhang type equations involving non-local "gradient terms". More precisely, let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial Ω$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form \[ \tag{KPZ} \left\{ \begin{aligned} (-Δ)^s u & = μ(x) |\mathbb{D}(u)|^q + λf(x), \quad && \mbox{ in } Ω,\\ u & = 0, && \mbox{ in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $q > 1$ and $λ> 0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $μ$ belongs to $L^{\infty}(Ω)$ and $\mathbb{D}$ represents a nonlocal "gradient term". Depending on the size of $λ> 0$, we derive existence and non-existence results. In particular, we solve several open problems posed in [4, Section 6] and [2, Section 7].

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arXiv2021

On the KPZ equation with fractional diffusion: global regularity and existence results

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-Δ)^s u &=&|\nabla u|^α+ f &\inn Ω_T\equivΩ\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminusΩ)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \innΩ,\\ \end{array}\right. $$ where $Ω$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2} \dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.

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arXiv2020

A note on quasilinear equations with fractional diffusion

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*} \left\lbrace \begin{array}{l} (-Δ)^{s}u + |\nabla u|^{p} =f \quad\text{ in } Ω \qquad \qquad \qquad \,\,\, u=0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus Ω, \quad s \in (1/2, 1). \end{array} \right. \end{eqnarray*} We are interested in the relation between the regularity of the source term $f$, and the regularity of the corresponding solution. If $p<2s$, that is the natural growth, we are able to show the existence for all $f\in L^1(Ø)$. In the subcritical case, that is, for $p < p_{*}:=N/(N-2s+1)$, we show that solutions are $\mathcal{C}^{1, α}$ for $f \in L^{m}$, with $m$ large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.

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arXiv2020

Towards a deterministic KPZ equation with fractional diffusion: The stationary problem

In this work we analyze the existence of solution to the fractional quasilinear problem, \begin{equation*} \left\{ \begin{array}{rcll} (-Δ)^s u &= & |\nabla u|^{p}+ łf & \text{ in }Ω, u &=& 0 &\hbox{ in } \mathbb{R}^N\setminusΩ, u&>&0 &\hbox{ in }Ω, \end{array}% \right. \end{equation*}% where $Ω\subset \ren$ is a bounded regular domain ($\mathcal{C}^2$ is sufficient), $s\in (\frac 12, 1)$, $1 2s$.

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arXiv2020

Fractional KPZ equations with critical growth in the gradient respect to Hardy potential

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=&λ\dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ μf &\inn Ω,\\ u&>&0 & \innΩ,\\ u&=&0 & \inn(\mathbb{R}^N\setminusΩ), \end{array}\right. $$ where $Ω$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s, μ>0$, $\frac{1}{2} 0$, there exists a critical exponent $p_{+}(λ, s)$ such that for $p> p_{+}(λ,s)$ there is no positive solution. Moreover, $p_{+}(λ,s)$ is optimal in the sense that, if $p<p_{+}(λ,s)$ there exists a positive solution for suitable data and $μ$ sufficiently small.

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arXiv2019

A note on the Fujita exponent in Fractional heat equation involving the Hardy potential

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-Δ)^s u=λ\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\ u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0 0$, $u_0\ge 0$, and $1<p<p_{+}(s,λ)$, where $p_{+}(λ, s)$ is the critical existence power found in \cite{BMP} and \cite{AMPP}.

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arXiv2018

Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications

Let $0 1$ be such that $ps 0$. 3- If $β\equiv \frac{N-ps}{2}$, as a consequence of the improved Hardy inequality, we obtain that for all $q<p$, there exists a positive constant $C(Ω)$ such that \begin{equation*} \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \dfrac{|u(x)-u(y)|^p}{|x-y|^{N+ps}|x|^β|y|^β} \,dy\,dx\ge C(Ω)\Big(\int_Ω \frac{|u(x)|^{p^*_{s,q}}}{|x|^{2β\frac{p^*_{s,q}}{p}}}\,dx\Big)^{\frac{p}{p^*_{s,q}}}, \end{equation*} for all $u\in \mathcal{C}^\infty_0(Ω)$ where $p^*_{s,q}=\frac{pN}{N-qs}$. \ Notice that the previous inequalities can be understood as the fractional extension of the Callarelli-Kohn-Nirenberg inequalities.

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arXiv2018

Nonlinear fractional Laplacian problems with nonlocal "gradient terms"

Let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-Δ)^s u & = μ(x)\, \mathbb{D}_s^{2}(u) + λf(x)\,, & \quad \mbox{in} Ω,\\ u & = 0\,, & \quad \mbox{in} \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $λ> 0$ is a real parameter, $f$ belongs to a suitable Lebesgue space, $μ\in L^{\infty}(Ω)$ and $\mathbb{D}_s^2$ is a nonlocal "gradient square" term given by \[ \mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2}\mbox{p.v.} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dy \,. \] Depending on the real parameter $λ> 0$, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón-Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

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arXiv2018

Nonlinear fractional elliptic problem with singular term at the boundary

Let $Ω\subset \mathbb{R}^N$ be a bounded regular domain, $0 2s$. We consider $$ (P)\left\{ \begin{array}{rcll} (-Δ)^s u &= & \frac{u^{q}}{d^{2s}} & \text{ in }Ω, \\ u &> & 0 & \text{in }Ω, \\ u & = & 0 & \text{ in }\mathbb{R}^N\setminusΩ,% \end{array}% \right. $$ where $0<q\le 2^*_s-1$, $0<s<1$ and $d(x) = dist(x,\partialΩ)$. {The main goal } of this paper is to analyze existence and non existence of solution to problem $(P)$ according to the value of $s$ and $q$.

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arXiv2017

Attainability of the fractional Hardy constant with nonlocal mixed boundary conditions. Applications

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \textit{fractional Hardy inequality } $$Λ_{N}\equivΛ_{N}(Ω):=\inf_{\{ϕ\in \mathbb{E}^s(Ω, D), ϕ\neq 0\}} \dfrac{\frac{a_{d,s}}{2} \displaystyle\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \dfrac{|ϕ(x)-ϕ(y)|^2}{|x-y|^{d+2s}}dx dy} {\displaystyle\int_Ω\frac{ϕ^2}{|x|^{2s}}\,dx}, $$ where $Ω$ is a bounded domain of $\mathbb{R}^d$, $0 & 0 &{\text{ in }} Ω, \mathcal{B}_{s}u&:=&uχ_{D}+\mathcal{N}_{s}uχ_{N}=0 &{\text{ in }}\mathbb{R}^{d}\backslash Ω, \\ \end{array}\right. $$ with $N$ and $D$ open sets in $\mathbb{R}^d\backslashΩ$ such that $N \cap D=\emptyset$ and $\overline{N}\cup \overline{D}= \mathbb{R}^d \backslashΩ$, $d>2s$, $λ> 0$ and $0<p\le 2_s^*-1$, $2_s^*=\frac{2d}{d-2s}$. We emphasize that the nonlinear term can be critical. The operators $(-Δ)^s $, fractional laplacian, and $\mathcal{N}_{s}$, nonlocal Neumann condition, are defined below in (1.5) and (1.6) respectively.

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arXiv2017

Existence of positive solutions to a nonlinear elliptic system with nonlinearity involving gradient term

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -Δu & = & v^q+\a g & \text{in }Ω, \\ -Δv& = &|\nabla u|^{p}+łf &\text{in }Ω, \\ u=v&=& 0 & \text{on }\partial Ω,\\ u,v& \geq & 0 & \text{in }Ω, \end{array}% \right. \end{equation*} where $Ω$ is a bounded domain of $\ren$ and $p\ge 1$, $q>0$ with $pq>1$. $f,g$ are nonnegative measurable functions with additional hypotheses and $\a, ł\ge 0$. As a consequence we show that the fourth order problem \begin{equation*} \left\{ \begin{array}{rcll} Δ^2 u & = &|\nabla u|^{p}+\tildeł \tilde{f} &\text{in }Ω, \\ u=\D u&=& 0 & \text{on }\partial Ω,\\ \end{array}% \right. \end{equation*} has a solution for all $p>1$, under suitable conditions on $\tilde{f}$ and $\tildeł$.

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arXiv2017

On fractional quasilinear parabolic problem with Hardy potential

The aim goal of this paper is to treat the following problem \begin{equation*} \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u &=&\dyle ł\dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } Ø_{T}=Ω\times (0,T), \\ u&\ge & 0 & \text{ in }\ren \times (0,T), \\ u &=& 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u(x,0)&=& u_0(x)& \mbox{ in }Ø, \end{array}% \right. \end{equation*} where $Ω$ is a bounded domain containing the origin, $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N, s\in (0,1)$ and $f, u_0$ are non negative functions. The main goal of this work is to discuss the existence of solution according to the values of $p$ and $ł$.

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arXiv2016

A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

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arXiv2016

On fractional p-laplacian parabolic problem with general data

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } Ø_{T}\equiv Ω\times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }Ø, \end{array}% \right. $$ where $Ω$ is a bounded domain, and $(-\D^s_{p})$ is the fractional p-Laplacian operator defined by $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N$, $s\in (0,1)$ and $f, u_0$ are measurable functions. The main goal of this work is to prove that if $(f,u_0)\in L^1(Ø_T)\times L^1(Ø)$, problem $(P)$ has a weak solution with suitable regularity. In addition, if $f_0, u_0$ are nonnegative, we show that the problem above has a nonnegative entropy solution. In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of $p$.

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