TY - JOUR
T1 - Isolated singularities in the heat equation behaving like fractional Brownian motions
AU - Fujii, Mikihiro
AU - Okada, Izumi
AU - Yanagida, Eiji
N1 - Funding Information:
MF was partly supported by Grant-in-Aid for JSPS Research Fellow (No. JP20J20941 ). IO was supported by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists (No. JP20K14329 ). EY was partly supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) (No. JP17H01095 ).
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/12/1
Y1 - 2021/12/1
N2 - We consider solutions of the linear heat equation in RN with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent α∈(0,1). We show that any isolated singularity is removable if it is weaker than a certain order depending on α. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent H∈(0,1/2]. It turns out that H=1/N is critical.
AB - We consider solutions of the linear heat equation in RN with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent α∈(0,1). We show that any isolated singularity is removable if it is weaker than a certain order depending on α. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent H∈(0,1/2]. It turns out that H=1/N is critical.
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U2 - 10.1016/j.jmaa.2021.125322
DO - 10.1016/j.jmaa.2021.125322
M3 - Article
AN - SCOPUS:85106392465
VL - 504
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
M1 - 125322
ER -