Harnack estimate for a semilinear parabolic equation

We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow- up of the positive solutions and recover a classical Harnack inequality. We also obtain a resu...

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Bibliographic Details
Published inScience China. Mathematics Vol. 60; no. 5; pp. 833 - 840
Main Authors Hou, SongBo, Zou, Liang
Format Journal Article
LanguageEnglish
Published Beijing Science China Press 01.05.2017
Springer Nature B.V
Subjects
Applications of Mathematics
Cauchy problems
Differential equations
Inequality
Mathematics
Mathematics and Statistics
不等式
估计
半线性抛物方程
微分
有限时间
柯西问题
椭圆型方程
正解
35K58
parabolic equation
finite-time blow-up
Harnack estimate
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Summary:We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow- up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.
Bibliography:We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow- up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.
11-5837/O1
parabolic equation, Harnack estimate, finite-time blow-up
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-016-0270-6