# On fractional quasilinear parabolic problem with Hardy potential

## TL;DR

This paper investigates the existence of solutions for a fractional quasilinear parabolic PDE with Hardy potential, considering different parameter regimes, contributing new insights into the interplay between fractional operators and singular potentials.

## Contribution

It provides new existence results for solutions of a fractional p-Laplacian parabolic problem with Hardy potential, depending on parameters p and λ.

## Key findings

- Existence of solutions varies with p and λ values.
- Conditions under which solutions exist are characterized.
- The work extends fractional PDE theory to include Hardy potentials.

## Abstract

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 \l \dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } \O_{T}=\Omega \times (0,T), \\ u&\ge & 0 & \text{ in }\ren \times (0,T), \\ u &=& 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u(x,0)&=& u_0(x)& \mbox{ in }\O, \end{array}% \right. \end{equation*} where $\Omega$ 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 $\l$.