Related papers: Probabilistic representation of solutions to the p…

The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(\gamma |\nabla u|^{p-2}\nabla u \right)…

That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous $p-$Laplace equation $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ are $C^{0,\alpha}$, for some $\alpha \in…

In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$…

In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla…

We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation…

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…

We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and…

The Dirichlet problem is considered both for degenerate and singular inhomogeneous quasilinear parabolic equations. We prove the existence of a solution $u$ such that $u_t$ belongs to $L_{\infty}$. The $L_{\infty}$ estimate of $u_t$ is…

We announce some new results for proving H\"older continuity of weak solutions to quasilinear parabolic equations whose prototype takes the form $$u_t - div (|\nabla u|^{p-2}\nabla u)= 0 \qquad \text{or} \qquad u_t - div…

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…

We study if two different solutions of the $p$-Laplace equation $$\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0,$$ where $1<p<\infty$, can coincide in an open subset of their common domain of definition. We obtain some partial results on this…

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1<p<2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay…

In this paper we present some non existence results concerning the stable solutions to the equation $$\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=g(x)f(u)\;\;\mbox{in}\;\;\mathbb{R}^N;\;\;p\geq 2$$ when $f(u)$ is either…

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…

We study the self-similar solutions of the equation \[ u_{t}-div(| \nabla u| ^{p-2}\nabla u)=0, \] in $\mathbb{R}^{N},$ when $p>2.$ We make a complete study of the existence and possible uniqueness of solutions of the form \[ u(x,t)=(\pm…

We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert…

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\nabla u$. More precisely, we prove that any locally bounded viscosity…

We study the uniqueness of non-negative solutions of the equation \begin{align*} \partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u). \end{align*} Basic estimates are derived with the Galerkin Method.

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-\Delta )^s$ for $s\in (\frac34,1)$, and we provide a new bootstrapping scheme that makes it possible to…

This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace…