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Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold. (English) Zbl 1448.58020

Summary: This paper determines when the Cauchy problem \[ \begin{cases} \partial_t u = \Delta u -Vu+ Wu^p \quad &\text{in } M \times (0, \infty ) \\ u(\cdot ,0)= u_0(\cdot ) \quad &\text{in } M \end{cases} \] 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. [K. Ishige, J. Math. Anal. Appl. 344, No. 1, 231–237 (2008; Zbl 1152.35051); R. Pinsky, J. Differ. Equations 246, No. 6, 2561–2576 (2009; Zbl 1162.35012); Q. S. Zhang, Duke Math. J. 97, No. 3, 515–539 (1999; Zbl 0954.35029); J. Differ. Equations 170, No. 1, 188–214 (2001; Zbl 0973.35035)]).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K10 Second-order parabolic equations
35B09 Positive solutions to PDEs
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