TY - JOUR
T1 - Weighted Lq(Lp)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives
AU - Han, Beom Seok
AU - Kim, Kyeong Hun
AU - Park, Daehan
N1 - Funding Information:
The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).
Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/8/5
Y1 - 2020/8/5
N2 - We present a weighted Lq(Lp)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂t αu(t,x)=Δu(t,x)+f(t,x),t>0,x∈Rd. Here, α∈(0,2) and ∂t α is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w1(x)∈Ap and w2(t)∈Aq, ∫0∞(∫Rd|uxx|pw1dx)q/pw2dt≤N∫0∞(∫Rd|f|pw1dx)q/pw2dt, where Ap is the class of Muckenhoupt Ap weights. Our approach is based on the sharp function estimates of the derivatives of solutions.
AB - We present a weighted Lq(Lp)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂t αu(t,x)=Δu(t,x)+f(t,x),t>0,x∈Rd. Here, α∈(0,2) and ∂t α is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w1(x)∈Ap and w2(t)∈Aq, ∫0∞(∫Rd|uxx|pw1dx)q/pw2dt≤N∫0∞(∫Rd|f|pw1dx)q/pw2dt, where Ap is the class of Muckenhoupt Ap weights. Our approach is based on the sharp function estimates of the derivatives of solutions.
KW - Caputo fractional derivative
KW - Fractional diffusion-wave equation
KW - L(L)-theory
KW - Muckenhoupt A weights
UR - http://www.scopus.com/inward/record.url?scp=85081203229&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2020.03.005
DO - 10.1016/j.jde.2020.03.005
M3 - Article
AN - SCOPUS:85081203229
SN - 0022-0396
VL - 269
SP - 3515
EP - 3550
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 4
ER -