# Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution equation with Neumann boundary conditions

Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution... The purpose of this paper is to investigate the time behavior of the solution of a weighted p-Laplacian evolution equation, given by \begin{aligned} {\left\{ \begin{array}{ll} u_{t} = \text {div} \left( \gamma |\nabla u|^{p-2}\nabla u \right) &{} \text {on } (0,\infty )\times S, \\ \gamma |\nabla u|^{p-2}\nabla u\cdot \eta =0 &{} \text {on } (0,\infty )\times \partial S, \\ u(0,\cdot )=u_{0} &{} \text {on } S,\end{array}\right. } \end{aligned} u t = div γ | ∇ u | p - 2 ∇ u on ( 0 , ∞ ) × S , γ | ∇ u | p - 2 ∇ u · η = 0 on ( 0 , ∞ ) × ∂ S , u ( 0 , · ) = u 0 on S , where $$n \in \mathbb {N}{\setminus } \{1\}$$ n ∈ N \ { 1 } , $$p \in (1,\infty ){\setminus } \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } , $$S\subseteq \mathbb {R}^{n}$$ S ⊆ R n is an open, bounded and connected set of class $$C^{1}$$ C 1 , $$\eta$$ η is the unit outer normal on $$\partial S$$ ∂ S , and $$\gamma :S\rightarrow (0,\infty )$$ γ : S → ( 0 , ∞ ) is a bounded function which can be extended to an $$A_{p}$$ A p -Muckenhoupt weight on $$\mathbb {R}^{n}$$ R n . It will be proven that the solution of (0.1) converges in $$L^{1}(S)$$ L 1 ( S ) to the average of the initial value $$u_{0} \in L^{1}(S)$$ u 0 ∈ L 1 ( S ) . Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Differential Equations and Applications NoDEA Springer Journals

# Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution equation with Neumann boundary conditions

, Volume 24 (4) – Jul 5, 2017
21 pages

/lp/springer_journal/asymptotic-results-for-solutions-of-a-weighted-varvec-p-p-laplacian-20vIxeSX1h
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
1021-9722
eISSN
1420-9004
D.O.I.
10.1007/s00030-017-0468-4
Publisher site
See Article on Publisher Site

### 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{aligned} {\left\{ \begin{array}{ll} u_{t} = \text {div} \left( \gamma |\nabla u|^{p-2}\nabla u \right) &{} \text {on } (0,\infty )\times S, \\ \gamma |\nabla u|^{p-2}\nabla u\cdot \eta =0 &{} \text {on } (0,\infty )\times \partial S, \\ u(0,\cdot )=u_{0} &{} \text {on } S,\end{array}\right. } \end{aligned} u t = div γ | ∇ u | p - 2 ∇ u on ( 0 , ∞ ) × S , γ | ∇ u | p - 2 ∇ u · η = 0 on ( 0 , ∞ ) × ∂ S , u ( 0 , · ) = u 0 on S , where $$n \in \mathbb {N}{\setminus } \{1\}$$ n ∈ N \ { 1 } , $$p \in (1,\infty ){\setminus } \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } , $$S\subseteq \mathbb {R}^{n}$$ S ⊆ R n is an open, bounded and connected set of class $$C^{1}$$ C 1 , $$\eta$$ η is the unit outer normal on $$\partial S$$ ∂ S , and $$\gamma :S\rightarrow (0,\infty )$$ γ : S → ( 0 , ∞ ) is a bounded function which can be extended to an $$A_{p}$$ A p -Muckenhoupt weight on $$\mathbb {R}^{n}$$ R n . It will be proven that the solution of (0.1) converges in $$L^{1}(S)$$ L 1 ( S ) to the average of the initial value $$u_{0} \in L^{1}(S)$$ u 0 ∈ L 1 ( S ) . Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.

### Journal

Nonlinear Differential Equations and Applications NoDEASpringer Journals

Published: Jul 5, 2017

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