Rachid Bentifour

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: ${\displaystyle M\left({\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}dxdy\right){(-\Delta )}^{s}u=\lambda \frac{u}{{\left|x\right|}^{2s}}+f(x,u,\nabla u)\mathrm{in}\Omega ,}$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain containing the origin, $s\in (0,1)$, $q\in (1,2]$ with $N>2s$, $\lambda >0$, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on $\lambda$. Two main cases areconsidered: $\left(I\right)-f(x,u,\nabla u)=u^p+\mu ,\mathrm{and}\left(II\right)-f(x,u,\nabla u)={|\nabla u|}^p+\mu g.$ Existence is proved under suitable hypotheses on $q,p$ and the data $g,\mu$. The results are new, including for the local case $s=1$.

The stability analysis of Timoshenko beam systems that incorporate delays and fractional boundary controls is a complex area of study in the field of viscoelasticity. Our study aims to balance the symmetric influence of internal viscoelastic damping and boundary fractional damping in a structured way. The goal is to establish a system where both effects contribute symmetrically to the overall stability and dynamics. In this paper, we study the stability of certain hyperbolic evolution problems, in particular, a Timoshenko system in viscoelasticity with fractional time delay and fractional boundary controls. We prove, under assumptions on the data, the lack of exponential stability decay rate when $\eta \ge 0$ and polynomial stability decay rate when $\eta >0$ using energy methods.

In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)

Let $0<s1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge 2$ For all $q<p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int_{{\mathbb R}^N} \frac{|u(x)|^p}{|x|^{p}}\,dx\geq C \int_{\Omega}\dint_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\mathbb R}^N)$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality. 2- Define $p^*_{s}=\frac{pN}{N-ps}$ and let $\beta<\frac{N-ps}{2}$, then \begin{equation*} \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}|x|^{\beta}|y|^{\beta}} \,dy\,dx\ge S(N,p,s,\beta)\Big(\int_{{\mathbb R}^N} \frac{|u(x)|^{p^*_{s}}}{|x|^{2\beta\frac{p^*_s}{p}}}\,dx\Big)^{\frac{p}{p^*_{s}}}, \end{equation*} for all $u\in \mathcal{C}^\infty_0(\Omega)$ where $S\equiv S(N,p,s,\beta)>0$. 3- If $\beta\equiv \frac{N-ps}{2}$, as a consequence of the improved Hardy inequality, we obtain that for all $q<p$, there exists a positive constant $C(\Omega)$ such that \begin{equation*} \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \dfrac{|u(x)-u(y)|^p}{|x-y|^{N+ps}|x|^{\beta}|y|^{\beta}} \,dy\,dx\ge C(\Omega)\Big(\int_{\Omega} \frac{|u(x)|^{p^*_{s,q}}}{|x|^{2\beta \frac{p^*_{s,q}}{p}}}\,dx\Big)^{\frac{p}{p^*_{s,q}}}, \end{equation*} for all $u\in \mathcal{C}^\infty_0(\Omega)$ 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.

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }\O, \end{array}% \right. $$ where $\Omega$ 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 $(f,u_0)\in L^1(\O_T)\times L^1(\O)$, 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 $p$.