Global boundedness for weak solutions of a class of time-fractional reaction-diffusion equations
The aim of this paper is to study in depth the global boundedness characterization of the weak solu-tions of fractional reaction-diffusion equations with respect to time and to compare them with traditional nonlo-cal reaction-diffusion models.In these fractional reaction-diffusion equations,we observe that the traditional first-order time derivative is replaced by the Caputo fractional derivative,thus introducing a new dimension to the time dynamics.We systematically explore the global boundedness characterization of the weak solutions by using a comprehensive approach.First,a differential inequality is constructed by using the Alikhanov in-equality and local energy estimates to verify the local boundedness of the solutions.A key aspect of this ap-proach is to fully utilize the asymptotic properties of Mittag-Leffler functions.Through in-depth study and analysis of these properties,we rigorously verify the local boundedness of the solution,which provides a ro-bust basis for further research on the global boundedness of solutions.In addition,the practical applications and theoretical significance of the fractional Duhamel formula are discussed in this paper.By flexibly applying the fractional Duhamel formula and its related properties to solve the equation,the local boundedness of the solution is extended to global boundedness.This approach not only opens a new track for the study of the na-ture of global boundedness of time-fractional order reaction-diffusion equations,but also reveals the intricate interplay between theoretical mathematics and complex dynamical systems,thereby further advancing cutting-edge research in the field.It should be emphasized that the proposed method not only adeptly addresses the in-herent computational barriers in the application of Duhamel's formula but also introduces a new perspective for explaining clearly the global boundedness feature of weak solutions.By synthesizing previous theoretical insights,this study provides a robust theoretical framework for scrutinizing inherently the complex dynamical applications in fractional reaction-diffusion equations in time.This theoretical and analytical approach signifi-cantly enriches our understanding of the behavior of such systems,thereby expanding the existing knowledge base in the field and promoting the progress of related research.
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