Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold
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Calculus of Variations
Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold Qingsong Gu1,2 · Yuhua Sun3 · Jie Xiao2 · Fanheng Xu4 Received: 23 December 2019 / Accepted: 31 July 2020 / Published online: 15 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper determines when the Cauchy problem
∂t u = u − V u + W u p u(·, 0) = u 0 (·)
in M × (0, ∞) in M
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). Mathematics Subject Classification Primary: 58J35 · Secondary: 35K10
Communicated by A. Malchiodi. 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).
B
Yuhua Sun [email protected] Qingsong Gu [email protected] Jie Xiao [email protected] Fanheng Xu [email protected]
1
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
2
Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada
3
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
4
School of Mathematics (Zhuhai), Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of China
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Q. Gu et al.
Contents 1 Introduction . . . . . . . . . . . . 1.1 Stating Theorem 1.1 . . . . . . 1.2 Transforming Theorem 1.1 . . 2 Verification . . . . . . . . . . . . . 2.1 Proving Theorem 1.1(i) . . . . 2.2 Proving Theorem 1.1(ii) . . . . 3 Illustration . . . . . . . . . . . . . 3.1 Exemplifying Theorem 1.1(i) . 3.2 Exemplifying Theorem 1.1(ii) References . . . . . . . . . . . . . . .
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1 Introduction 1.1 Stating Theorem 1.1 Working on a connected non-compact geodesically complete Riemannian manifold
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