arXiv 0802.18602008-02-14math.APmath.DS

Allen-Cahn方程吸引子维数的一个精确估计和变换

A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations

Nikos I. Karachalios, Nikos B. Zographopoulos

备注：: 9 pages

摘要：

考虑半线性反应扩散方程$\partial_tϕ-νΔϕ-V(x)ϕ+f(ϕ)=0$，$ν>0$在有界区域$Ω\subset\mathbb{R}^N$上.我们假设标准的Allen-Cahn型非线性，而势$V$要么是平方反比势$V(x)=δ|x|^{-2}$，要么是边界势$V(x)=δ\mathrm{dist}(x,\partialΩ)^{-2}$，$δ\geq 0$（因此包括经典的Allen-Cahn方程作为$δ=0$的特殊情况）。在次临界情形$δ=0$，$N\geq 1$和$0<μ:=\fracδν<μ^*$，$N\geq 3$（其中$μ^*$是Hardy和Hardy型不等式的最优常数）下，给出了全局吸引子维数的一个新估计.这个估计是由A. D. Melas（Proc. Amer. Math. Soc. \textbf{131}（2003），631-636）。估计是尖锐的，揭示了存在（明确给定）阈值的体积的比例$Ω$的惯性矩上的吸引子的尺寸可能会有很大的变化。考虑到有限维的全局吸引子在临界情况下$μ=μ^*$。

英文摘要：

We consider the semilinear reaction diffusion equation $\partial_tϕ-νΔϕ-V(x)ϕ+f(ϕ)=0$, $ν>0$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$ is either the inverse square potential $V(x)=δ|x|^{-2}$ or the borderline potential $V(x)=δ\mathrm{dist}(x,\partialΩ)^{-2}$, $δ\geq 0$ (thus including the classical Allen-Cahn equation as a special case when $δ=0$). In the subcritical cases $δ=0$, $N\geq 1$ and $0<μ:=\fracδν<μ^*$, $N\geq 3$ (where $μ^*$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new estimate on the dimension of the global attractor. This estimate comes out by an improved lower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003), 631-636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio of the volume to the moment of inertia of $Ω$ on which the dimension of the attractor may considerably change. Consideration is also given on the finite dimensionality of the global attractor in the critical case $μ=μ^*$.

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