Related papers: p-Laplacian Keller-Segel Equation: Fair Competitio…

We investigate the diffusion-aggregation equations with degenerate diffusion $\Delta u^m$ and singular interaction kernel $\mathcal{K}_s = (-\Delta)^{-s}$ with $s\in(0,\frac{d}{2})$. We analyze the regime %($m>2-2s/d$, $d$ is the dimension)…

We consider a nonlocal aggregation diffusion equation incorporating repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction. When the attractive interaction kernel is radially symmetric and strictly…

The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the…

We investigate the following fractional $p$-Laplacian equation \[ \begin{cases} \begin{aligned} (-\Delta)_p^s u&=\lambda |u|^{q-2}u+|u|^{p_s^*-2}u &&\text{in}~\Omega,\\ u &=0 &&\text{in}~ \mathbb{R}^n\setminus\Omega, \end{aligned}…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard…

This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the…

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the…

We consider reaction-diffusion equations driven by the $p$-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $L^2$ spectrum bounded away from zero, the main example we…

For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies H\"older boundedness for…

This paper considers the Dirichlet problem $$ -\mathrm{div}(a\nabla u_a)=f \quad \hbox{on}\,\,\ D, \qquad u_a=0\quad \hbox{on}\,\,\partial D, $$ for a Lipschitz domain $D\subset \mathbb R^d$, where $a$ is a scalar diffusion function. For a…

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and…

This paper deals with the homogenization of the $p$-Laplacian reaction-diffusion problems in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of pure-reactive type. We generalize…

In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &\rho _t=\nabla \cdot \left( \left| \nabla \rho \right|^{p-2}\nabla \rho \right) -\chi \nabla \cdot \left(…

In this paper, we consider the existence of solutions of the following nonhomogeneous fractional $p(x,.)$-Laplacian Dirichlet problem: \begin{equation*} \left\{\begin{aligned} \Big(-\Delta_{p(x,.)}\Big)^s u (x)&=f(x, u) &\text { in }&…

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration \[…

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a…

We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic…

In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the…

In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem: \begin{equation*} \begin{cases} M\left([u]^p_{p,s_1}\right)(-\Delta)^{s_1}_p u +…