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2024 Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source
Bo Liang, Qingchun Li, Yongbo Zhu, Yongzheng Zhu
Author Affiliations +
Taiwanese J. Math. Advance Publication 1-22 (2024). DOI: 10.11650/tjm/240404

Abstract

The existence and asymptotic behavior of solutions a fourth-order partial differential equation with a $p$-Laplacian diffusion and a nonlinear source are studied by using potential well theory. When the initial functionals satisfy $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \gt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \geq 0$, the existence and exponential decay result of weak solutions are given. For $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \lt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \lt 0$, we obtain the blow-up behavior at a finite time for weak solutions. For $\mathcal{F}(w_{0}) \gt d$, we show the global existence for small initial datum and blow-up for big initial datum. Moreover, the uniqueness holds for bounded solutions. In addition, we show that the $p$-Laplacian term has an essential effect to the source function so that we add some growth conditions to $g(w)$.

Funding Statement

This work is supported by the Education Department Science Foundation of Liaoning Province of China (No. LJKMZ20220832) and the Research Start-up Fund of Chuzhou University (No. 2024).

Citation

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Bo Liang. Qingchun Li. Yongbo Zhu. Yongzheng Zhu. "Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source." Taiwanese J. Math. Advance Publication 1 - 22, 2024. https://doi.org/10.11650/tjm/240404

Information

Published: 2024
First available in Project Euclid: 7 May 2024

Digital Object Identifier: 10.11650/tjm/240404

Subjects:
Primary: 35G20 , 35K25 , 35K35

Keywords: blow up behavior , fourth-order partial differential equations , global solutions , potential well

Rights: Copyright © 2024 The Mathematical Society of the Republic of China

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