图上的莱本森方程
Leibenson's equation on graphs
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中文总结 AI 辅助
本文在无限图上研究莱本森方程，证明其全局解的存在唯一性，结合Faber-Krahn不等式得到平滑估计与传播界，在特定图上确定衰减率与灭绝二分性。
中文摘要 AI 辅助
本文在无限图上研究莱本森方程$$ \partial_t u = \Delta_p u^q, $$其中$p>1$，$q>0$，$\Delta_p$表示离散$p$-拉普拉斯算子。我们证明，对于任意可积初始数据$u_0$，存在全局解，且在$p$和$q$的某一范围内该解是唯一的。假设存在Faber-Krahn不等式，我们得到了精确的$\ell^1$-$\ell^\infty$平滑估计，以及初始支撑有限的解的传播定量界。在$p$和$q$满足某些假设时，我们还证明了当图满足等周不等式时解的有限时间灭绝结果。特别地，在多项式体积增长的Cayley图上，当$q(p-1)>1$时，我们确定了非负有限质量解的$\ell^\infty$范数的最优长时间衰减率，并证明了关于耗尽解有限时间灭绝的精确二分性。
英文摘要
In this paper we study on infinite graphs the Leibenson equation $$ \partial_t u = Δ_p u^q, $$ where $p>1$, $q>0$ and $Δ_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber--Krahn inequality, we obtain sharp $\ell^1$-$\ell^\infty$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an \textit{isoperimetric inequality}. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the $\ell^\infty$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions.
