arXiv subjects

Rachid Bentifour

Publications and source records attributed to Rachid Bentifour.

3 recordsLinked to original sources

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.

math.AP↗

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 $ł$.

math.AP↗

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 $p$.

math.AP↗