Asymptotic Results for Solutions of a weighted p-Laplacian evolution Equation with Neumann Boundary Conditions

Abstract

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(γ|\nabla u|^{p-2}\nabla u \right) & \text{on} (0,\infty)\times S, \\ γ|\nabla u|^{p-2}\nabla u\cdotη=0 & \text{on} (0,\infty)\times \partial S, \\ u(0,\cdot)=u_{0} & \text{on} S,\end{cases} \end{align} where $n\in \mathbb{N}\setminus \{1\}$, $p\in (1,\infty )\setminus \{2\}$, $S\subseteq {\mathbb{R}}^n$ is an open, bounded and connected set of class $C^1$, $\eta$ is the unit outer normal on $\partial S$, and $\gamma :S\to (0,\infty )$ is a bounded function which can be extended to an $A_p$-Muckenhoupt weight on ${\mathbb{R}}^n$. It will be proven that the solution converges in $L^1(S)$ to the average of the initial value $u_0\in L^1(S)$. Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.

The final paper is published in NoDEA (August 2017 - Volume 24 Issue 4) at http://link.springer.com

References in corpus (1)

• Mathematical Models for Erosion and the Optimal Transportation of Sediment

Cited by in corpus (1)

• A randomized weighted $p$-Laplacian evolution equation with Neumann boundary conditions