拟线性热方程的尖锐衰减率与渐近简化

中文摘要 AI 辅助

我们研究形如 $ \dot u-\Delta u = \operatorname{div B}(\nabla u)$ 的拟线性热方程的衰减率。右端项被解释为对普通热方程的非线性扰动，我们将证明解的最优衰减率。更精确地，在算子 $B \colon \R^d \to \R^d$ 的自然条件下，我们证明对于足够正则的初始数据，解在大时间下按 $t^{-d/4}$ 衰减，其梯度按 $t^{-d/4 - 1/2}$ 衰减，即与自由热方程相同的速率。此外，我们证明拟线性解与线性解之差严格地比任一方程的解通常的衰减更快，即存在渐近简化。所有估计都是完全显式的，并依赖于最初由 Schonbek 引入的傅里叶分裂技巧的变体。

英文摘要

We study the decay rate for a quasilinear heat equation of the form $ \dot u-Δu = \operatorname{div B}(\nabla u)$. The term on the right-hand-side is interpreted as a nonlinear perturbation of the ordinary heat equation and we will prove an optimal decay rate for the solution. More precisely, under natural conditions on the operator $B \colon \R^d \to \R^d$, we show that for sufficiently regular initial data the solution decays as $t^{-d/4}$ and its gradient as $t^{-d/4 - 1/2}$ for large times, that is at the same rates as for the free heat equation. Furthermore, we prove that the difference between the quasilinear and the linear solution decays strictly faster than the solution of either equation generically does, that is that there is asymptotic simplification. All estimates are completely explicit and rely on variations of Fourier splitting techniques, originally introduced by Schonbek.

发表机构

• Normandie University, INSA de Rouen Normandie（诺曼底大学，鲁昂高等师范学院）

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