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The Blow-Up of Solutions for a Class of Semi-linear Equations with p-Laplacian Viscoelastic Term Under Positive Initial Energy

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Abstract

This paper deals with homogeneous Dirichlet boundary value problem to a class of semi-linear equations with p-Laplacian viscoelastic term

$$\begin{aligned} \frac{\partial u}{\partial t}-\Delta u+\int _{0}^{t}g(t-s)\Delta _{p}u(x,s){{\textrm{d}}}s=\left| u\right| ^{q(x)-2}u,\quad x\in \Omega ,\ t\ge 0, \end{aligned}$$

the bounded domain \(\Omega \subset R^{n}~(n\ge 3)\) with a smooth boundary. We prove that the weak solutions of the above problems blow up in finite time for all \(2k<q^-<q^+<p<\frac{2n}{n-2}\) (k is defined in (2.5)), when the initial energy is positive and the function g satisfies suitable conditions. This result generalized and improved the result by Messaoudi (Abstr Appl Anal 2005(2):87–94, 2005).

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Acknowledgements

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions.

Funding

This work is supported by the National Natural Science Foundation of China (12171054) and the Natural Science Foundation of Jilin Province, Free Exploration Basic Research. (YDZJ202201ZYTS584).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun, 130022, People’s Republic of China

    Xiulan Wu, Xiaoxin Yang & Yaxin Zhao

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  1. Xiulan Wu
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  2. Xiaoxin Yang
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  3. Yaxin Zhao
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XW, XY and YZ wrote the main manuscript text. All the authors reviewed the manuscript.

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Correspondence to Xiulan Wu.

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Wu, X., Yang, X. & Zhao, Y. The Blow-Up of Solutions for a Class of Semi-linear Equations with p-Laplacian Viscoelastic Term Under Positive Initial Energy. Mediterr. J. Math. 20, 272 (2023). https://doi.org/10.1007/s00009-023-02440-z

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