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Dynamic Properties of the p-Laplacian Reaction–Diffusion Equation in Multi-dimensional Space

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Abstract

In this paper we study the p-Laplacian reaction–diffusion equation

$$\begin{aligned} u_{t}-\text{ div }(|\nabla u|^{p-2}\nabla u)=k(t)f(u) \end{aligned}$$

subject to appropriate initial and boundary conditions. We show the positive solution \(u(\pmb {x},t )\) exists globally, under the conditions on f, k and the boundary conduction function. It is proved that the solution blows up at finite time, for some initial data and additional energy type conditions, by establishing accurate estimates and using the Sobolev inequality in multi-dimensional space.

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Acknowledgements

The authors thank the editor and anonymous referees for their valuable suggestions and comments, which improved the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, People’s Republic of China

    Shuai Zheng & Fushan Li

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  1. Shuai Zheng
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  2. Fushan Li
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The authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

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Correspondence to Fushan Li.

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This work was supported by Natural Science Foundation of Shandong Province of China (ZR2019MA067, ZR2020MA018)

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Zheng, S., Li, F. Dynamic Properties of the p-Laplacian Reaction–Diffusion Equation in Multi-dimensional Space. Qual. Theory Dyn. Syst. 20, 53 (2021). https://doi.org/10.1007/s12346-021-00494-6

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