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Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

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Abstract

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation

$${u_t} - div\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) = - {\left| u \right|^{\beta - 1}}u + \alpha {\left| u \right|^{q - 2}}u,$$

where p > 1; β > 0, q ≥ 1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11371286 and 11401458), the Special Fund of Education Department (Grant No. 2013JK0586) and the Youth Natural Science Grant of Shaanxi Province of China (Grant No. 2013JQ1015).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Yan Li & Zhengce Zhang

  2. College of Science, Xi’an University of Architecture & Technology, Xi’an, 710055, China

    Liping Zhu

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  1. Yan Li
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  2. Zhengce Zhang
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  3. Liping Zhu
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Correspondence to Zhengce Zhang.

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Li, Y., Zhang, Z. & Zhu, L. Classification of certain qualitative properties of solutions for the quasilinear parabolic equations. Sci. China Math. 61, 855–868 (2018). https://doi.org/10.1007/s11425-016-9077-8

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