arXiv 2005.047842020-05math.AP

具有双稳态反应项的 $p$-Laplacian 扩散问题解的长时间动力学

Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms

Raffaele Folino, Ramón G. Plaza, Marta Strani

备注：: 29 pages, 5 figures
发表信息：: Discrete Contin. Dyn. Syst., 41 (2021), 3211-3240
DOI：: 10.3934/dcds.2020403

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AI总结：

研究带双势阱反应项的 $p$-Laplacian 扩散方程，通过 Ginzburg-Landau 型有效能量界刻画过渡层在临界、超临界与亚临界情形下的长时间动力学。

摘要：

本文建立了 $p$-Laplacian（非线性）演化方程缓慢移动过渡层解的出现性，该方程为 $u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u)$，其中 $x\in(a,b)$、$t>0$，$\varepsilon>0$ 和 $p>1$ 为常数；该现象由一族双势阱势 $F(u)=\frac{1}{2n}|1-u^2|^n$ 的作用驱动，参数 $n>1$、$n\in\mathbb{R}$，势在两个纯相 $u=\pm1$ 处取得极小值。方程配有初始条件和 Neumann 型边界条件。结果表明：界面层，即初始时除有限个宽度为 $\varepsilon$ 的薄过渡区外均等于 $\pm1$ 的解，在临界情形 $n=p$ 下持续指数长时间，在超临界（或退化）情形 $n>p$ 下持续代数长时间。为此，文中建立了一类重整化的 Ginzburg-Landau 型有效能量势的能量界。相比之下，在亚临界情形 $n<p$ 下，过渡层解是稳态的。

英文摘要：

This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$ and $p>1$ are constants, driven by the action of a family of double-well potentials of the form \[ F(u)=\frac{1}{2n} |1-u^2|^{n}, \] indexed by $n>1$, $n\in\mathbb{R}$ with minima at two pure phases $u = \pm 1$. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$, persist for an exponentially long time in the critical case with $n=p$, and for an algebraically long time in the supercritical (or degenerate) case with $n > p$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are established. In contrast, in the subcritical case with $n<p$, the transition layer solutions are stationary.

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