We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved, in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson's Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation's solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

中文翻译：

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非退化非线性扩散方程的长时间渐近

我们研究了原型非线性扩散方程的长时间渐近性。具体来说，我们考虑非退化扩散函数的情况，该函数是问题的因变量的（非负）多项式。我们使用爱因斯坦的随机游走范式来激发这些类型的方程，导致非发散形式的偏微分方程。另一方面，使用守恒原理会导致发散形式的偏微分方程。导出一个转换来处理这两种情况。然后，证明了一个最大值原理（在无界域和有界域上），以获得非线性扩散问题解的时间演化的上下界。具体来说，这些界限是基于线性问题的基本解（所谓的 Aronson' s 格林函数）。将非线性问题的解的长时间渐近性夹在线性问题的两个基本解之间，我们证明，与退化扩散的情况不同，非退化扩散方程的解收敛到线性扩散解在很长时间。选择数值例子支持数学定理并说明收敛过程。我们的结果对如何解释应用物理文献中讨论过的潜在异常扩散过程（例如在颗粒材料的流动中）的渐近标度具有影响。与退化扩散的情况不同，非退化扩散方程的解长时间收敛到线性扩散解。选择数值例子支持数学定理并说明收敛过程。我们的结果对如何解释应用物理文献中讨论过的潜在异常扩散过程（例如在颗粒材料的流动中）的渐近标度具有影响。与退化扩散的情况不同，非退化扩散方程的解长时间收敛到线性扩散解。选择数值例子支持数学定理并说明收敛过程。我们的结果对如何解释应用物理文献中讨论过的潜在异常扩散过程（例如在颗粒材料的流动中）的渐近标度具有影响。