System of degenerate parabolic p-Laplacian
Sung‐Hoon Kim, Ki-Ahm Lee · Open Mathematics · 2024
Abstract In this article, we study the mathematical properties of the solution u = ( u 1 , … , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) to the degenerate parabolic system u t = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) , ( p > 2 ) . {{\bf{u}}}_{t}= abla \hspace{0.25em}\cdot \hspace{0.25em}({| abla {\bf{u}}| }^{p-2} abla {\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 u l {u}^{l} , ( 1 ≤ l ≤ k ) (1\le l\le k) , converges to the function c l ℬ {c}^{l}{\mathcal{ {\mathcal B} }} in space as t →</