Decay rates and initial values for time-fractional diffusion-wave equations

2021/03/10 by Masahiro Yamamoto, Yamamoto, Masahiro
Mathematics · #35B40 #35C20 #35R11 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2103.06013

openalex publication_date 2021/03/10 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28

Abstract

We consider a solution u(⋅,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α∈ (0,2) ∖ \ 1\ where t is a time variable. We first prove that a suitable norm of u(⋅,t) is bounded by (1)/(t^α) for 00. Moreover we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α=1, the decay rate can give some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.

Citations

• Fractional Differential Equations

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