论文标题
$ p $ - 拉普拉斯式扩散问题的长期动力与可反应术语
Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms
论文作者
论文摘要
本文确定了$ p $ -laplacian（nonlinear）进化方程的缓慢移动过渡层解决方案的出现，\ [u_t = \ varepsilon^p（| u_x |^|^{p-2} u_x）_x-_x-x-_x-f'（u_x） - f'（u），f'（u），\ qquad x \ in（a，a，b） t> 0，\]其中$ \ varepsilon> 0 $和$ p> 1 $是常数，这是由形式的双孔潜力的动作驱动的，\ [f（u）= \ frac {1} {2n} {2n} | 1-u^2 |^{n}，\ n}，\ n $ n> 1 $ n> $ n> 1 $ n> inim，inim $ n> 1 $ n \ in。纯阶段$ u = \ pm 1 $。该方程赋予了Neumann类型的初始条件和边界条件。结果表明，界面层或最初等于$ \ pm 1 $的解决方案除外，除了有限的宽度宽度$ \ varepsilon $，在$ n = p $的关键情况下，持续很长一段时间，并且在超级策略（或dementareate）的代数（或dementareate）的情况下持续了长时间的代数。为此，建立了金茨堡 - 兰豪类型的重新归一化有效能量潜力的能量界限。相反，在$ n <p $的亚临界情况下，过渡层解决方案是固定的。
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.
