论文标题
通过退化扩散率的通用Allen-CAHN模型的解决方案的长期行为
Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity
论文作者
论文摘要
广义的Allen-Cahn方程， \ [ u_t = \ varepsilon^2（d（u）u_x）_x- \ frac {\ varepsilon^2} 2d'（u）u_x^2-f'（u）， \] 随着非线性扩散，$ d = d（u）$和潜力，$ f = f（u）$的表格 \ [ d（u）= | 1-u^2 |^{m}，\ quad \ text {或} \ quad d（u）= | 1-u |^{m}，\ quad m> 1， \] 和 \ [ f（u）= \ frac {1} {2n} | 1-u^2 |^{n}，\ qquad n \ geq2， \] 分别研究了。这些选择对应于一个可以从双重井电位中得出的反应函数，并且根据一个或在两个井中消失的密度$ u $，$ u = \ pm 1 $的密度$ u $。结果表明，接口层解决方案等于$ \ pm 1 $，除非在有限数量的宽度宽度$ \ varepsilon $上持续长时间或代数长的时间，这取决于指数$ n $ n $和$ m $之间的相互作用。为此，得出了金茨堡 - 兰道类型的重新归一化有效能量潜力的能量界限。
The generalized Allen-Cahn equation, \[ u_t=\varepsilon^2(D(u)u_x)_x-\frac{\varepsilon^2}2D'(u)u_x^2-F'(u), \] with nonlinear diffusion, $D = D(u)$, and potential, $F = F(u)$, of the form \[ D(u) = |1-u^2|^{m}, \quad \text{or} \quad D(u) = |1-u|^{m}, \quad m >1, \] and \[ F(u)=\frac{1}{2n}|1-u^2|^{n}, \qquad n\geq2, \] respectively, is studied. These choices correspond to a reaction function that can be derived from a double well potential, and to a generalized degenerate diffusivity coefficient depending on the density $u$ that vanishes at one or at the two wells, $u = \pm 1$. It is shown that interface layer solutions that are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$ persist for an either exponentially or algebraically long time, depending upon the interplay between the exponents $n$ and $m$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are derived.
