American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential
  • DCDS Home
  • This Issue
  • Next Article
    Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems
doi: 10.3934/dcds.2021002

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Beom-Seok Han , Kyeong-Hun Kim and  Daehan Park ,

Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea

* Corresponding author

Received  February 2020 Revised  November 2020 Published  January 2021

Fund Project: The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324)

We introduce a weighted
$ L_{p} $
-theory (
$ p>1 $
) for the time-fractional diffusion-wave equation of the type
$ \begin{equation*} \partial_{t}^{\alpha}u(t,x) = a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x), \quad t>0, x\in \Omega, \end{equation*} $
where
$ \alpha\in (0,2) $
,
$ \partial_{t}^{\alpha} $
 denotes the Caputo fractional derivative of order
$ \alpha $
, and
$ \Omega $
 is a
$ C^1 $
 domain in
$ \mathbb{R}^d $
. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.
Keywords: Time-fractional equation, Caputo fractional derivative, Sobolev space with weights, variable coefficients, C1 domains.
Mathematics Subject Classification: 45D05, 45K05, 45N05, 35B65, 26A33.
Citation: Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021002
References:
[1]

D. Beleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216.  Google Scholar

[2]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020), 108338.  doi: 10.1016/j.jfa.2019.108338.  Google Scholar

[3]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.  Google Scholar

[5]

B.-S. Han, K.-H. Kim and D. Park, Weighed $L_{q}(L_{p})$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives, J. Differ. Equ., 269 (2020), 3515-3550.  doi: 10.1016/j.jde.2020.03.005.  Google Scholar

[6]

D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). Google Scholar

[7]

I. Kim, K.-H. Kim and S. Lim, A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives, Ann. Probab., 47 (2019), 2087-2139.  doi: 10.1214/18-AOP1303.  Google Scholar

[8]

I. Kim, K.-H. Kim and S. Lim, An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.  Google Scholar

[9]

K.-H. Kim, On $L_{p}$-theory of stochastic partial differential equations of divergence form in $C^{1}$ domains, Probab. Theory. Rel., 130 (2004), 473-492.  doi: 10.1007/s00440-004-0368-5.  Google Scholar

[10]

K.-H. Kim, On stochastic partial differential equations with variable coefficients in $C^1$ domains, Stoch. Proc. Appl., 112 (2004), 261-283.  doi: 10.1016/j.spa.2004.02.006.  Google Scholar

[11]

K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.  doi: 10.1137/S0036141003421145.  Google Scholar

[12]

K.-H. Kim and N. V. Krylov, On SPDEs with variable coefficients in one space dimension, Potential Anal., 21 (2004), 209-239.  doi: 10.1023/B:POTA.0000033334.06990.9d.  Google Scholar

[13]

K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325.  doi: 10.1137/S0036141097326908.  Google Scholar

[15]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008. doi: 10.1090/gsm/096.  Google Scholar

[16]

N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Comm. Partial Differential Equations, 24 (1999), 1611-1653.  doi: 10.1080/03605309908821478.  Google Scholar

[17]

G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329.  Google Scholar

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods. Appl. Anal., 7 (2000), 195-204.  doi: 10.4310/MAA.2000.v7.n1.a9.  Google Scholar

[19]

F. Mainardi, Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97.   Google Scholar

[20]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar

[21]

R. Metzler, E. Barkai and J. Kalfter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 35-63.   Google Scholar

[22]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[23]

I. Podludni, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[24]

H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. Google Scholar

[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993.  Google Scholar

[26]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[27]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[28]

R. Zacher, Global strong solvability of a quasilinear subdiffusion problem, J. Evol. Equ., 12 (2012), 813-831.  doi: 10.1007/s00028-012-0156-0.  Google Scholar

show all references

References:
[1]

D. Beleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216.  Google Scholar

[2]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020), 108338.  doi: 10.1016/j.jfa.2019.108338.  Google Scholar

[3]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.  Google Scholar

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.  Google Scholar

[5]

B.-S. Han, K.-H. Kim and D. Park, Weighed $L_{q}(L_{p})$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives, J. Differ. Equ., 269 (2020), 3515-3550.  doi: 10.1016/j.jde.2020.03.005.  Google Scholar

[6]

D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). Google Scholar

[7]

I. Kim, K.-H. Kim and S. Lim, A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives, Ann. Probab., 47 (2019), 2087-2139.  doi: 10.1214/18-AOP1303.  Google Scholar

[8]

I. Kim, K.-H. Kim and S. Lim, An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.  Google Scholar

[9]

K.-H. Kim, On $L_{p}$-theory of stochastic partial differential equations of divergence form in $C^{1}$ domains, Probab. Theory. Rel., 130 (2004), 473-492.  doi: 10.1007/s00440-004-0368-5.  Google Scholar

[10]

K.-H. Kim, On stochastic partial differential equations with variable coefficients in $C^1$ domains, Stoch. Proc. Appl., 112 (2004), 261-283.  doi: 10.1016/j.spa.2004.02.006.  Google Scholar

[11]

K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.  doi: 10.1137/S0036141003421145.  Google Scholar

[12]

K.-H. Kim and N. V. Krylov, On SPDEs with variable coefficients in one space dimension, Potential Anal., 21 (2004), 209-239.  doi: 10.1023/B:POTA.0000033334.06990.9d.  Google Scholar

[13]

K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[14]

N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298-325.  doi: 10.1137/S0036141097326908.  Google Scholar

[15]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008. doi: 10.1090/gsm/096.  Google Scholar

[16]

N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Comm. Partial Differential Equations, 24 (1999), 1611-1653.  doi: 10.1080/03605309908821478.  Google Scholar

[17]

G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329.  Google Scholar

[18]

S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods. Appl. Anal., 7 (2000), 195-204.  doi: 10.4310/MAA.2000.v7.n1.a9.  Google Scholar

[19]

F. Mainardi, Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97.   Google Scholar

[20]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar

[21]

R. Metzler, E. Barkai and J. Kalfter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 35-63.   Google Scholar

[22]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[23]

I. Podludni, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[24]

H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. Google Scholar

[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993.  Google Scholar

[26]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[27]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[28]

R. Zacher, Global strong solvability of a quasilinear subdiffusion problem, J. Evol. Equ., 12 (2012), 813-831.  doi: 10.1007/s00028-012-0156-0.  Google Scholar

[1]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[2]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[3]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
[4]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019
[5]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[6]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[7]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[8]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
[9]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[10]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[11]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[12]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017
[13]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[14]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[15]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
[16]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[17]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[18]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[19]

Todd Hurst, Volker Rehbock. Optimizing micro-algae production in a raceway pond with variable depth. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021027
[20]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (17)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences