Liouville type theorem for a parabolic equation involving Pucci operator

Liu Yong

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Liouville type theorem for a parabolic equation involving Pucci operator

Liu Yong * #

Department of Mathematics and Physics, North China Electric Power University,Beijing,102206

*Correspondence author

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Funding: National Natural Science Foundation of China （No.No. 11101141）

Opened online:17 October 2012

Accepted by: none

Citation: Liu Yong.Liouville type theorem for a parabolic equation involving Pucci operator[OL]. [17 October 2012] http://en.paper.edu.cn/en_releasepaper/content/4491338

The main subject of this paper is the superlinear parabolic equation involving Pucci's operator. Due to the fully nonlinear property of this operator, it is one of the main concerns in the Pde theory. One natural question is whether or not the results for the Laplacian operator has analogies for Pucci's operator. While this is in general the case for elliptic equation, the main purpose of this paper is investigating the similarity between these two operators in the parabolic eqauation catagory. One classicla result for the superlinear parabolic equation involving Laplacian operator is the Liouville type theorem. Due to the importance of these kind of theorems, the Liouville type theorem for parabolic equation involving Pucci's operator is studied in this paper. When the space variable is one dimensional, it is relatively easy to analyze the Pucci operator. The main result of this paper states that in this case, the corresponding equation does not have global positive bounded solution. This result could be used to get certain a priori estimates for a class of fully nonlinear equations.

Keywords:Partial differential equation; Parabolic equation; Pucci operator; Global solution; Liouville type theorem

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