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On the existence with exponential decay and the blow-up of solutions for coupled systems of semi-linear corner-degenerate parabolic equations with singular potentials

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Abstract

In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincaré inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hua Chen

  2. School of Science, Wuhan University of Technology, Wuhan, 430070, China

    Nian Liu

Authors
  1. Hua Chen
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  2. Nian Liu
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Correspondence to Nian Liu.

Additional information

This work was supported by NFSC (11871387 and 11931012).

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Chen, H., Liu, N. On the existence with exponential decay and the blow-up of solutions for coupled systems of semi-linear corner-degenerate parabolic equations with singular potentials. Acta Math Sci 41, 257–282 (2021). https://doi.org/10.1007/s10473-021-0115-3

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  • DOI: https://doi.org/10.1007/s10473-021-0115-3

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