Masahiro Yamamoto†
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 t1-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
†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 ¨, Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania 4 Correspondence Member of Accademia Peloritana dei Pericolanti, Palazzo Universita, 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.
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 16 no 1, 2024
