TY - JOUR

T1 - Classical unique continuation property for multi-term time-fractional evolution equations

AU - Lin, Ching Lung

AU - Nakamura, Gen

N1 - Funding Information:
The authors thank the anonymous reviewers who gave several useful comments which help us a lot improving our paper. As for the funding, the first author was partially supported by the Ministry of Science and Technology of Taiwan. Also, the second author acknowledge the support by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (No.15H05740) during this study.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/2

Y1 - 2023/2

N2 - As for the unique continuation property (UCP) of solutions in (0 , T) × Ω with a domain Ω⊂Rn,n∈N for a multi-terms time fractional evolution equation, we have already shown it by assuming that the solutions are zero for t≤ 0 (see [12]). Here the strongly elliptic operator for this evolution equation can depend on time and the orders of its time-fractional derivatives are in (0, 2). This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for t≤ 0. We have achieved this aim by first using the usual Holmgren transformation together with the argument in [12] to derive the UCP in (T1, T2) × B1 for some 0 < T1< T2< T and a ball B1⊂ Ω. Then if u is the solution of the equation with u= 0 in (T1, T2) × B1, we show u= 0 also in ((0 , T1] ∪ [T2, T)) × Br for some r< 1 by using the result and argument in [12] which use two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time-fractional evolution equation.

AB - As for the unique continuation property (UCP) of solutions in (0 , T) × Ω with a domain Ω⊂Rn,n∈N for a multi-terms time fractional evolution equation, we have already shown it by assuming that the solutions are zero for t≤ 0 (see [12]). Here the strongly elliptic operator for this evolution equation can depend on time and the orders of its time-fractional derivatives are in (0, 2). This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for t≤ 0. We have achieved this aim by first using the usual Holmgren transformation together with the argument in [12] to derive the UCP in (T1, T2) × B1 for some 0 < T1< T2< T and a ball B1⊂ Ω. Then if u is the solution of the equation with u= 0 in (T1, T2) × B1, we show u= 0 also in ((0 , T1] ∪ [T2, T)) × Br for some r< 1 by using the result and argument in [12] which use two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time-fractional evolution equation.

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U2 - 10.1007/s00208-021-02341-0

DO - 10.1007/s00208-021-02341-0

M3 - Article

AN - SCOPUS:85123104780

SN - 0025-5831

VL - 385

SP - 551

EP - 574

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -