DECAY RATES AND INITIAL VALUES FOR TIME-FRACTIONAL DIFFUSION-WAVE EQUATIONS*

Masahiro Yamamoto[†]

Dedicated to Dr. Dan Tiba on the occasion of his 70th anniversary

Abstract. We consider a solution u(•, t) to an inițial boundary value problem for time-fractional diffusion-wave equation with the order a G (0,2) \{ 1} where t is a time variable. We first prove that a suitable norm of u(•, t) is bounded by the rate t^-a for 0 < a < 1 and t^1-a for 1 < a < 2 for all large t > 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation a = 1, the decay rate can keep 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.

MSC: R11, 35B40, 35C20

Keywords: fractional diffusion-wave equation, decay rate, initial value

DOI 10.56082/annalsarscimath.2024.1.77

Abstract Article Volume 16 no 1 / 2024

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*Accepted for publication on November 24-th, 2023

[†]myama@ms.u-tokyo.ac.jp ¹ Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan ² Zonguldak Bulent Ecevit University, Turkey ³ Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania ⁴ Corre- spondence Member of Accademia Peloritana dei Pericolanti, Palazzo Universită, Piazza S. Pugliatti 1 98122 Messina, Italy. Paper written with financial supports of Grant-in-Aid for Scientific Research (A) 20H00117 and Grant-in-Aid for Challenging Research (Pioneering) 21K18142 of Japan Society for the Promotion of Science.