含$u/ε$依赖的粘性次线性Hamilton--Jacobi方程的均匀化
Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/ε$-Dependence
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中文总结 AI 辅助
本文研究含未知函数快速依赖的粘性次线性Hamilton--Jacobi方程的周期均匀化，证明小参数下的定性均匀化结果并建立演化胞腔问题的大时间平均结论。
中文摘要 AI 辅助
我们研究一类含未知函数快速依赖项的粘性Hamilton--Jacobi方程的周期均匀化问题。该问题兼具带$u^ε/ε$周期哈密顿量的一阶Hamilton--Jacobi方程，与带快速振荡正势的半线性热方程的特征。本文证明了当$|η|$足够小时，$H(y,s,p)=F(s,p)+ηW(y,s,p)$的定性均匀化结果，其小阈值依赖于初值的Lipschitz常数，并建立了一类一般演化胞腔问题的大时间平均结果。
英文摘要
We study the periodic homogenization of a class of viscous Hamilton--Jacobi equations with fast dependence on the unknown. This problem combines features of first-order Hamilton--Jacobi equations with \(u^ε/ε\)-periodic Hamiltonians and semilinear heat equations with rapidly oscillating positive potentials. In this paper, we prove qualitative homogenization results for $H(y,s,p)=F(s,p)+ηW(y,s,p)$ when $|η|$ is sufficiently small, with the smallness threshold depending on the Lipschitz constant of the initial datum, and establish a large-time averaging result for a general class of evolutionary cell problems.
