论文标题
与基质值电位相连的线性抛物线系统的平衡收敛
Convergence to equilibrium for linear parabolic systems coupled by matrix-valued potentials
论文作者
论文摘要
我们考虑抛物线线性方程的系统，但要受$ \ Mathbb {r}^d $中有界域的neumann边界条件的影响，这些系统由矩阵价值的电位$ v $耦合，并在哪些条件下调查了该系统的每个条件在哪些条件下，该系统对$ t \ t \ t \ t \ iffty $ contion to to to to to to to fore formibirium。 尽管这显然是关于抛物线方程系统的一个基本问题，但直到现在，仅在对潜在$ v $的一定积极假设下进行了研究。没有积极性，就无法应用Perron-Frobenius理论，并且该问题似乎是敞开的。在本文中，我们解决了所有$ \ ell^p $ dipsipative in [1，\ infty] $中的$ p $的潜力。虽然CASE $ P = 2 $可以通过经典的Hilbert Space方法来处理，但对于$ p \ not = 2 $，此事变得更加精致。我们通过采用最近与$ l^p $ - 空间的几何结构密切相关的光谱理论结果来解决这个问题。
We consider systems of parabolic linear equations, subject to Neumann boundary conditions on bounded domains in $\mathbb{R}^d$, that are coupled by a matrix-valued potential $V$, and investigate under which conditions each solution to such a system converges to an equilibrium as $t \to \infty$. While this is clearly a fundamental question about systems of parabolic equations, it has been studied, up to now, only under certain positivity assumptions on the potential $V$. Without positivity, Perron-Frobenius theory cannot be applied and the problem is seemingly wide open. In the present article, we address this problem for all potentials that are $\ell^p$-dissipative for some $p \in [1,\infty]$. While the case $p=2$ can be treated by classical Hilbert space methods, the matter becomes more delicate for $p \not= 2$. We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of $L^p$-spaces.
