Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 89, pp. 1-21.

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

Authors: Fanmeng Meng (Anhui Univ., Hefei, China)
         Xian-Feng Zhou (Anhui Univ., Hefei, China)

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.