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

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. For more information see https://ejde.math.txstate.edu/Volumes/2025/89/abstr.html