Searcharxiv⌕ Search

arXiv subjects

Guillaume Warnault

Publications and source records attributed to Guillaume Warnault.

9 recordsLinked to original sources
arXiv2026

Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-Δ)_p^s u_λ=λu_λ^q - b(x)u_λ^r \quad \text{in} \;Ω,\\ u_λ=0 \quad \text{in} \; ( \mathbb{R}^d \backslash Ω), \\ u_λ>0 \text{ in} \; Ω. \end{cases}\ \end{equation*} We also prove behavior of $u_λ$ with respect to $λ,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.

math.AP↗
arXiv2024

Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium

In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t u+(-Δ)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times Ω,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash Ω, \\ u(0,\cdot)=u_0 & \text{in} \; Ω. \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization.

math.AP↗
arXiv2021

Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: \begin{center} $\begin{cases} Δ_p u= v^m| \nabla u |^α& \text{in}\quad Ω\\ Δ_p v= v^β| \nabla u |^q & \text{in}\quad Ω, \end{cases}$ \end{center} where $Ω\subset\mathbb{R}^N(N\geq 2)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1 0$, $α\geq 0$, $0\leq β\leq m$ and $δ:=(p-1-α)(p-1-β)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that,we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\R^3$.

math.AP↗
arXiv2020

Regularity results for a class of nonlinear fractional Laplacian and singular problems

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(Ω)$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \leq δ< 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $δ\geq sp.$ Moreover, by virtue of barrier arguments we study the behavior of minimal weak solution in terms of distance function. Consequently, we prove Hölder regularity up to the boundary and optimal Sobolev regularity for minimal weak solutions.

math.AP↗
arXiv2019

Doubly nonlinear equation involving $p(x)$-homogeneous operators: local existence, uniqueness and global behaviour

In this work, we investigate the qualitative properties as uniqueness, regularity and stabilization of the weak solution to the nonlinear parabolic problem involving general $p(x)$-homogeneous operators: \begin{equation*} \left\{ \begin{alignedat}{2} {} \frac{q}{2q-1}\partial_t(u^{2q-1}) -\nabla.\, a(x, \nabla u) & {}= f(x,u) + h(t,x) u^{q-1} && \quad\mbox{ in } \, (0,T) \times Ω; u & {}> 0 && \quad\mbox{ in }\, (0,T) \times Ω; u & {}= 0 && \quad\mbox{ on }\, (0,T) \times \partialΩ; u(0,.)&{}= u_0 && \quad\ \mbox{in}\, \ Ω. \end{alignedat} \right. \end{equation*} Thanks to the Picone's identity obtained in [10], we prove new results about comparison principles which yield a priori estimates, positivity and uniqueness of weak solutions.

math.AP↗
arXiv2018

A picone Identity for variable exponent operators and applications

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the $p(x)$-Laplacian defined as $\mbox{div}(|\nabla u|^{p(x)-2} \nabla u).$ Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

math.AP↗
arXiv2012

Entire large solutions for semilinear elliptic equations

We analyze the semilinear elliptic equation $Δu=ρ(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $ρ$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.

math.AP↗