# American Institute of Mathematical Sciences

July  2021, 41(7): 3415-3445. doi: 10.3934/dcds.2021002

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

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

* Corresponding author

Received  February 2020 Revised  November 2020 Published  July 2021 Early access  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.
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 and Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. 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. [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. [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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [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. [6] D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). [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. [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. [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. [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. [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. [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. [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. [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. [15] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008. doi: 10.1090/gsm/096. [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. [17] G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329. [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. [19] F. Mainardi, Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97. [20] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [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. [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. [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. [24] H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. [25] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993. [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. [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. [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.

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##### 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. [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. [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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [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. [6] D. Kim, K.-H. Kim and K. Lee, Parabolic systems with measurable coefficients in weighted Sobolev spaces, arXiv: 1809.01325, (2018). [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. [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. [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. [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. [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. [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. [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. [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. [15] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, 2008. doi: 10.1090/gsm/096. [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. [17] G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329. [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. [19] F. Mainardi, Fractional diffusive waves in viscoelastic solids, Nonlinear waves in solids, 137 (1995), 93-97. [20] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust., 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [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. [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. [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. [24] H. Richard, Fractional Calculus: An Introduction for Physicists, , World Scientific, 2014. [25] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, , CRC Press, 1993. [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. [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. [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.
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