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Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold

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Abstract

This paper determines when the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} {{\partial _t u} = \Delta u -Vu+ Wu^p} &{}\quad \text{ in } M \times (0, \infty ) \\ {u(\cdot ,0)= {u_0(\cdot )}} &{}\quad \text{ in } M \end{array} \right. \end{aligned}$$

has no global positive solution on a connected non-compact geodesically complete Riemannian manifold for a given triple (V, W, p). As the principal result of this paper, Theorem 1.1 optimally extends in a unified way most of the previous results in this subject (cf. Ishige in J Math Anal Appl 344:231–237, 2008; Pinsky in J Differ Equ 246(6):2561–2576, 2009; Zhang in Duke Math J 97:515–539, 1999; Zhang in J Differ Equ 170:188–214, 2001).

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Acknowledgements

The authors would like to not only express their deep gratitude to Qi S. Zhang who initiated a study of the Cauchy problem presented in this paper in connection with his paper [27], but also thank A. Grigor’yan & I. Verbitsky for their helpful communications. Moreover, the authors are very grateful to the anonymous referee for many valuable comments improving the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Qingsong Gu

  2. Department of Mathematics and Statistics, Memorial University, St. John’s, NL, A1C 5S7, Canada

    Qingsong Gu & Jie Xiao

  3. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Yuhua Sun

  4. School of Mathematics (Zhuhai), Sun Yat-Sen University (Zhuhai Campus), Zhuhai, 519082, People’s Republic of China

    Fanheng Xu

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  1. Qingsong Gu
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Correspondence to Yuhua Sun.

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Communicated by A. Malchiodi.

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Yuhua Sun (+ Fanheng Xu) was supported by the National Natural Science Foundation of China (#11501303 - #11871296) and Tianjin Natural Science Foundation (#19JCQNJC14600); Jie Xiao (+ Qingsong Gu) was supported by NSERC of Canada (#20171864).

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Gu, Q., Sun, Y., Xiao, J. et al. Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold. Calc. Var. 59, 170 (2020). https://doi.org/10.1007/s00526-020-01837-y

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