变系数半线性抛物方程的梯度估计与刘维尔定理
Gradient estimate and Liouville theorem for a semilinear parabolic equation with variable coefficient
浏览论文内容
中文总结 AI 辅助
本文针对完备黎曼流形上带变系数的半线性抛物方程，通过Nash-Moser迭代建立正解的Li-Yau型梯度估计，进而推导古老解与永恒解的刘维尔定理，推广了相关经典结果。
中文摘要 AI 辅助
本文研究了完备黎曼流形上的半线性抛物方程∂_t f−Δf=a(x,t)f^p，该流形的里奇曲率有下界。通过Nash-Moser迭代，我们为该方程正解建立了Li-Yau型梯度估计，其中系数函数a可以是严格定号或变号的。作为应用，我们在非负里奇曲率的流形上推导了古老解与永恒解的刘维尔定理，推广了一系列经典结果。我们的证明特点是在辅助量中引入并调整两个参数，以适配变系数a并扩展p的容许范围。
英文摘要
In this paper, we investigate the semilinear parabolic equation $\partial_t f-Δf=a(x,t)f^p$ on a complete Riemannian manifold with Ricci curvature bounded from below. By means of Nash-Moser iteration, we establish Li-Yau type gradient estimates for positive solutions to this equation, where the coefficient function $a$ can be either strictly sign-definite or sign-changing. As an application, we derive Liouville theorems for ancient and eternal solutions on manifolds with nonnegative Ricci curvature, generalizing a number of classical results. Our proof features the incorporation and tuning of two parameters in the auxiliary quantities to accommodate the variable coefficient $a$ and to extend the admissible range of $p$.
