Open Mathematics (Nov 2024)
System of degenerate parabolic p-Laplacian
Abstract
In this article, we study the mathematical properties of the solution u=(u1,…,uk){\bf{u}}=({u}^{1},\ldots ,{u}^{k}) to the degenerate parabolic system ut=∇⋅(∣∇u∣p−2∇u),(p>2).{{\bf{u}}}_{t}=\nabla \hspace{0.25em}\cdot \hspace{0.25em}({| \nabla {\bf{u}}| }^{p-2}\nabla {\bf{u}}),\hspace{1.0em}(p\gt 2). More precisely, we show the existence and uniqueness of solution u{\bf{u}} and investigate a priori L∞{L}^{\infty } boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component ul{u}^{l}, (1≤l≤k)(1\le l\le k), converges to the function clℬ{c}^{l}{\mathcal{ {\mathcal B} }} in space as t→∞t\to \infty . Here, the function ℬ{\mathcal{ {\mathcal B} }} is the fundamental or Barenblatt solution of pp-Laplacian equation, and the constant cl{c}^{l} is determined by the L1{L}^{1}-mass of ul{u}^{l}. The proof is based on the existence of entropy functional. As an application of the asymptotic large-time behaviour, we establish a Harnack-type inequality, which makes the size of the spatial average controlled by the value of the solution at one point.
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