Electron. J. Differential Equations, Vol. 2025 (2025), No. 89, pp. 1-21.

Decay estimates and extinction properties of parabolic equations with classical and fractional time derivatives

Fanmeng Meng, Xian-Feng Zhou

Abstract:
In this article, we study the decay estimates and extinction properties of weak solutions to some parabolic equations with classical and fractional time derivatives. Firstly, we establish a new comparison principle for parabolic equations with mixed time derivatives. Based on this comparison principle and energy methods, we obtain the power-law decay estimates for weak solutions of nonhomogeneous abstract parabolic problems with mixed time-derivatives. Furthermore, we present three specific applications of the decay results for the abstract parabolic problem. Finally, we discus the finite time extinction property of the weak solution for the 1-Kirchhoff type parabolic problem with mixed time-derivatives.

Submitted July 1, 2025. Published September 16, 2025.
Math Subject Classifications: 35B40, 26A33, 35K90.
Key Words: Abstract parabolic equation; Caputo derivative; decay estimates; extinction properties; comparison principle.
DOI: 10.58997/ejde.2025.89

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Fanmeng Meng
School of Mathematical Sciences
Anhui University
Hefei 230601, China
email: mengfanmeng96@163.com
Xian-Feng Zhou
School of Mathematical Sciences
Anhui University
Hefei 230601, China
email: zhouxf@ahu.edu.cn

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