We study the large time behavior of solutions $v:\Omega\times(0,\infty)\rightarrow \mathbb{R}$ of the PDE $\partial_t(|v|^{p-2}v)=\Delta_pv.$ We show that $e^{\left(\lambda_p/(p-1)\right)t}v(x,t)$ converges to an extremal of a Poincare inequality on $\Omega$ with optimal constant $\lambda_p$, as $t\rightarrow \infty$. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincare inequality on $\Omega$. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

中文翻译：

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特鲁丁格方程解的大时间行为

我们研究了 PDE $\partial_t(|v|^{p-2}v)=\Delta_pv 的解 $v:\Omega\times(0,\infty)\rightarrow \mathbb{R}$ 的大时间行为.$ 我们证明 $e^{\left(\lambda_p/(p-1)\right)t}v(x,t)$ 收敛于 $\Omega$ 上的 Poincare 不等式的极值，具有最优常数 $\ lambda_p$，如 $t\rightarrow \infty$。我们还证明了解的大时间值近似于 $\Omega$ 上相应的“双”庞加莱不等式的极值。此外，我们的理论使我们能够推导出涉及各种边界条件和非局部算子的相关双非线性流的大时间渐近线。