TY - JOUR

T1 - Asymptotic expansions of solutions of fractional diffusion equations

AU - Ishige, Kazuhiro

AU - Kawakami, Tatsuki

AU - Michihisa, Hironori

N1 - Funding Information:
The first author was supported by Grant-in-Aid for Scientific Research (A) (15H02058) from the Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Young Scientists (B) (16K17629) from the Japan Society for the Promotion of Science.
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

PY - 2017

Y1 - 2017

N2 - In this paper we obtain the precise description of the asymptotic behavior of the solution u of the fractional diffusion equation ∂tu + (-Δ)θ/2 u = 0 in RN × (0,∞) with the initial data φ ∈ LK := L1(RN, (1 + |x|)K dx), where 0 < θ < 2 and K ≥ 0. This enables us to obtain the asymptotic behavior of the hot spots of the solution u. Furthermore, we develop the arguments in [K. Ishige and T. Kawakami, Math. Ann., 353 (2012), pp. 161-192] and [K. Ishige, T. Kawakami, and K. Kobayashi, J. Evol. Equ., 14 (2014), pp. 749-777] and establish a method to obtain the asymptotic expansions of the solutions to inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

AB - In this paper we obtain the precise description of the asymptotic behavior of the solution u of the fractional diffusion equation ∂tu + (-Δ)θ/2 u = 0 in RN × (0,∞) with the initial data φ ∈ LK := L1(RN, (1 + |x|)K dx), where 0 < θ < 2 and K ≥ 0. This enables us to obtain the asymptotic behavior of the hot spots of the solution u. Furthermore, we develop the arguments in [K. Ishige and T. Kawakami, Math. Ann., 353 (2012), pp. 161-192] and [K. Ishige, T. Kawakami, and K. Kobayashi, J. Evol. Equ., 14 (2014), pp. 749-777] and establish a method to obtain the asymptotic expansions of the solutions to inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

KW - Anomalous diffusion

KW - Asymptotic expansion

KW - Fractional diffusion equation

KW - Hot spot

KW - Semilinear parabolic equation

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U2 - 10.1137/16M1101428

DO - 10.1137/16M1101428

M3 - Article

AN - SCOPUS:85021731107

VL - 49

SP - 2167

EP - 2190

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 3

ER -