半线性热传导方程Cauchy问题生命跨度估计  

Lifespan Estimate for the Cauchy Problem of Semilinear Heat Equations

作　　者：林银河[1] 徐根海[1] 杨姗姗 LIN Yinhe;XU Genhai;YANG Shanshan(l.Faculty of Engineering,Lishui University,Lishui 323000,Zhejiang;School of Science,Zhejiang Sci-Tech University,Hangzhou 310018,Zhejiang)

机构地区：[1]丽水学院工学院,浙江丽水323000 [2]浙江理工大学理学院,浙江杭州310018

出　　处：《丽水学院学报》2020年第5期1-7,共7页Journal of Lishui University

基　　金：浙江省自然科学基金资助项目“带变系数耗散项的半线性波动方程解的大时间行为”(LY18A010008)。

摘　　要：众所周知,Rn中的热核具有半群性质,小初值半线性热传导方程的柯西问题存在临界指标,即Fujita临界指标p=pF(n)=1+2/n。如果1<p≤pF,则解将在有限时间内破裂,而如果p>pF,则存在整体解。因此,在破裂情形,有必要估计解的生命跨度。在已有生命跨度下界估计的基础上,文章利用热核的半群性质,用一种比较简单的方法建立了解的生命跨度上界估计,从而验证了生命跨度估计的最优性。It is well known that the heat kernel in R^n has semigroup property, and the Cauchy problem of semilinear heat equation with small data admits a critical power, i.e. p=p_F(n)=1+2/n, which is the so called Fujita power. The solution will blow up in a finite time if 1<p≤p_F, while it exists globally in time if p>p_F. Thus, in the blow up case we need to estimate the lifespan of the solution. In this paper, based on the lower bound of the lifespan estimate, we focus on the upper bound and prove the sharpness of the lifespan estimate, the upper bound estimation of life span estimation is established by a simple method, which verifies the optimality of lifespan estimation by using the semigroup property of the heat kernel.

关 键 词：生命跨度 半线性热传导方程 半群 柯西问题 

分 类 号：O175.2[理学—数学]