TY - JOUR
T1 - An Lq(Lp)-theory for diffusion equations with space-time nonlocal operators
AU - Kim, Kyeong Hun
AU - Park, Daehan
AU - Ryu, Junhee
N1 - Funding Information:
The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354 ).
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/6/25
Y1 - 2021/6/25
N2 - We present an Lq(Lp)-theory for the equation ∂tαu=ϕ(Δ)u+f,t>0,x∈Rd;u(0,⋅)=u0. Here p,q>1, α∈(0,1), ∂tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ0∈(0,1] and c>0 such that [Formula presented] We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove ‖|∂tαu|+|u|+|ϕ(Δ)u|‖Lq([0,T];Lp)≤N(‖f‖Lq([0,T];Lp)+‖u0‖Bp,qϕ,2−2/αq), where Bp,qϕ,2−2/αq is a modified Besov space on Rd related to ϕ. Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for p≠q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.
AB - We present an Lq(Lp)-theory for the equation ∂tαu=ϕ(Δ)u+f,t>0,x∈Rd;u(0,⋅)=u0. Here p,q>1, α∈(0,1), ∂tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ0∈(0,1] and c>0 such that [Formula presented] We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove ‖|∂tαu|+|u|+|ϕ(Δ)u|‖Lq([0,T];Lp)≤N(‖f‖Lq([0,T];Lp)+‖u0‖Bp,qϕ,2−2/αq), where Bp,qϕ,2−2/αq is a modified Besov space on Rd related to ϕ. Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for p≠q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.
KW - Caputo fractional derivative
KW - Integro-differential operator
KW - L(L)-theory
KW - Space-time nonlocal equations
UR - http://www.scopus.com/inward/record.url?scp=85103757116&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2021.04.003
DO - 10.1016/j.jde.2021.04.003
M3 - Article
AN - SCOPUS:85103757116
VL - 287
SP - 376
EP - 427
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
ER -