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December  2021, 14(12): 4551-4574. doi: 10.3934/dcdss.2021113

Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

1. 

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

2. 

Department of Mathematics and Computer Sciences, University of Science, (Viet Nam National University), Ho Chi Minh City, Viet Nam

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied $ L^p-L^q $ estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case $ 0<{\alpha}<1 $ and $ {\alpha} = 1 $. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between $ L^p $ spaces and Hilbert scales spaces.

Citation: Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4551-4574. doi: 10.3934/dcdss.2021113
References:
[1]

Y. E. AghdamH. SafdariY. AzariH. Jafari and D. Baleanu, Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2025-2039.  doi: 10.3934/dcdss.2020402.  Google Scholar

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.  Google Scholar

[3]

A. AlsaediB. AhmadM. Kirane and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.  doi: 10.1515/anona-2020-0153.  Google Scholar

[4]

V. V. Au, J. Singh and A. T. Nguyen, Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients, Electronic Research Archive, (2021). doi: 10.3934/era.2021052.  Google Scholar

[5]

N. T. BaoT. CaraballoN. H. Tuan and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity, 34 (2021), 1448-1502.  doi: 10.1088/1361-6544/abc4d9.  Google Scholar

[6]

T. CaraballoT. B. NgocN. H. Tuan and R. Wang, On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffler kernel, Proc. Amer. Math. Soc., 149 (2021), 3317-3334.  doi: 10.1090/proc/15472.  Google Scholar

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Y. ChenH. GaoM. J. Garrido-Atienza and B. Schmalfu\ss, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.  Google Scholar

[8]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[9]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with a logarithmic potential, Milan Journal of Mathematics, 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[10]

P. Colli and T. Fukao, Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type, Adv. Nonlinear Anal., 9 (2020), 16-38.  doi: 10.1515/anona-2018-0055.  Google Scholar

[11]

E. CuestaM. KiraneA. Alsaedi and B. Ahmad, On the sub-diffusion fractional initial value problem with time variable order, Adv. Nonlinear Anal., 10 (2021), 1301-1315.  doi: 10.1515/anona-2020-0182.  Google Scholar

[12]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in ${\mathbb R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.  Google Scholar

[13]

R. M. GanjiH. JafariN. S. Nkomo and S. P. Moshokoa, A mathematical model and numerical solution for brain tumor derived using fractional operator, Results in Physics, 28 (2021), 104671.   Google Scholar

[14]

M. GrasselliG. SchimpernaA. Segatti and S. Zelik, On the 3D Cahn–Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.  doi: 10.1007/s00028-009-0017-7.  Google Scholar

[15]

M. GrasselliG. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737.  doi: 10.1088/0951-7715/23/3/016.  Google Scholar

[16]

J. HanR. Xu and Y. Yang, Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation, Asymptotic Analysis, 122 (2021), 349-369.  doi: 10.3233/ASY-201621.  Google Scholar

[17]

K. IshigeN. Miyake and S. Okabe, Blowup for a fourth-order parabolic equation with gradient nonlinearity, SIAM J. Math. Anal., 52 (2020), 927-953.  doi: 10.1137/19M1253654.  Google Scholar

[18]

A. Iuorio and S. Melchionna, Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction, Discrete Contin. Dyn. Syst., 38 (2018), 3765-3788.  doi: 10.3934/dcds.2018163.  Google Scholar

[19]

H. JafariR. M. GanjiN. S. Nkomo and Y. P. Lv, A numerical study of fractional order population dynamics model, Results in Physics, 27 (2021), 104456.   Google Scholar

[20]

M. Krasnoschok, V. Pata, S. V. Siryk and N. Vasylyeva, A subdiffusive Navier-Stokes-Voigt system, Phys. D, 409 (2020), 132503, 13 pp. doi: 10.1016/j.physd.2020.132503.  Google Scholar

[21]

C. Li and Z. Li, Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3659-3683.  doi: 10.3934/dcdss.2021023.  Google Scholar

[22]

L. Li and G.-J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.  Google Scholar

[23]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[24]

S. LiuF. Wang and H. Zhao, Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.  doi: 10.1016/j.jde.2007.02.014.  Google Scholar

[25]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.  Google Scholar

[26]

A. Miranville, The Cahn-Hilliard equation with a nonlinear source term, J. Differential Equations, 294, 88–117. doi: 10.1016/j.jde.2021.05.045.  Google Scholar

[27]

A. Segatti and J. L. Vázquez, On a fractional thin film equation, Adv. Nonlinear Anal., 9 (2020), 1516-1558.  doi: 10.1515/anona-2020-0065.  Google Scholar

[28]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[29]

H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2017. Google Scholar

[30]

N. H. TuanV. V. Au and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Commun. Pure Appl. Anal., 20 (2021), 583-621.  doi: 10.3934/cpaa.2020282.  Google Scholar

[31]

N. H. Tuan and T. Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations, Proc. Amer. Math. Soc., 149 (2021), 143-161.  doi: 10.1090/proc/15131.  Google Scholar

[32]

N. H. Tuan and Y. Zhou, Well-posedness of an initial value problem for fractional diffusion equation with Caputo-Fabrizio derivative, J. Comput. Appl. Math., 375 (2020), 112811, 21 pp. doi: 10.1016/j.cam.2020.112811.  Google Scholar

[33]

R. XuT. ChenC. Liu and Y. Ding, Global well-posedness and global attractor of fourth order semilinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 1515-1529.  doi: 10.1002/mma.3165.  Google Scholar

[34]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Applied Numerical Mathematics, 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[35]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.  Google Scholar

[37]

H. YeQ. Liu and Z.-M. Chen, Global existence of solutions of the time fractional Cahn–Hilliard equation in $\mathbb R^3$, J. Evol. Equ., 21 (2021), 2377-2411.  doi: 10.1007/s00028-021-00687-1.  Google Scholar

[38]

X. Zheng, H. Wang and H. Fu, Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative, Chaos Solitons Fractals, 138 (2020), 109966, 7 pp. doi: 10.1016/j.chaos.2020.109966.  Google Scholar

show all references

References:
[1]

Y. E. AghdamH. SafdariY. AzariH. Jafari and D. Baleanu, Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2025-2039.  doi: 10.3934/dcdss.2020402.  Google Scholar

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.  Google Scholar

[3]

A. AlsaediB. AhmadM. Kirane and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.  doi: 10.1515/anona-2020-0153.  Google Scholar

[4]

V. V. Au, J. Singh and A. T. Nguyen, Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients, Electronic Research Archive, (2021). doi: 10.3934/era.2021052.  Google Scholar

[5]

N. T. BaoT. CaraballoN. H. Tuan and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity, 34 (2021), 1448-1502.  doi: 10.1088/1361-6544/abc4d9.  Google Scholar

[6]

T. CaraballoT. B. NgocN. H. Tuan and R. Wang, On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffler kernel, Proc. Amer. Math. Soc., 149 (2021), 3317-3334.  doi: 10.1090/proc/15472.  Google Scholar

[7]

Y. ChenH. GaoM. J. Garrido-Atienza and B. Schmalfu\ss, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.  Google Scholar

[8]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[9]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with a logarithmic potential, Milan Journal of Mathematics, 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[10]

P. Colli and T. Fukao, Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type, Adv. Nonlinear Anal., 9 (2020), 16-38.  doi: 10.1515/anona-2018-0055.  Google Scholar

[11]

E. CuestaM. KiraneA. Alsaedi and B. Ahmad, On the sub-diffusion fractional initial value problem with time variable order, Adv. Nonlinear Anal., 10 (2021), 1301-1315.  doi: 10.1515/anona-2020-0182.  Google Scholar

[12]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in ${\mathbb R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.  Google Scholar

[13]

R. M. GanjiH. JafariN. S. Nkomo and S. P. Moshokoa, A mathematical model and numerical solution for brain tumor derived using fractional operator, Results in Physics, 28 (2021), 104671.   Google Scholar

[14]

M. GrasselliG. SchimpernaA. Segatti and S. Zelik, On the 3D Cahn–Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.  doi: 10.1007/s00028-009-0017-7.  Google Scholar

[15]

M. GrasselliG. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737.  doi: 10.1088/0951-7715/23/3/016.  Google Scholar

[16]

J. HanR. Xu and Y. Yang, Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation, Asymptotic Analysis, 122 (2021), 349-369.  doi: 10.3233/ASY-201621.  Google Scholar

[17]

K. IshigeN. Miyake and S. Okabe, Blowup for a fourth-order parabolic equation with gradient nonlinearity, SIAM J. Math. Anal., 52 (2020), 927-953.  doi: 10.1137/19M1253654.  Google Scholar

[18]

A. Iuorio and S. Melchionna, Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction, Discrete Contin. Dyn. Syst., 38 (2018), 3765-3788.  doi: 10.3934/dcds.2018163.  Google Scholar

[19]

H. JafariR. M. GanjiN. S. Nkomo and Y. P. Lv, A numerical study of fractional order population dynamics model, Results in Physics, 27 (2021), 104456.   Google Scholar

[20]

M. Krasnoschok, V. Pata, S. V. Siryk and N. Vasylyeva, A subdiffusive Navier-Stokes-Voigt system, Phys. D, 409 (2020), 132503, 13 pp. doi: 10.1016/j.physd.2020.132503.  Google Scholar

[21]

C. Li and Z. Li, Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3659-3683.  doi: 10.3934/dcdss.2021023.  Google Scholar

[22]

L. Li and G.-J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.  Google Scholar

[23]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[24]

S. LiuF. Wang and H. Zhao, Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.  doi: 10.1016/j.jde.2007.02.014.  Google Scholar

[25]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.  Google Scholar

[26]

A. Miranville, The Cahn-Hilliard equation with a nonlinear source term, J. Differential Equations, 294, 88–117. doi: 10.1016/j.jde.2021.05.045.  Google Scholar

[27]

A. Segatti and J. L. Vázquez, On a fractional thin film equation, Adv. Nonlinear Anal., 9 (2020), 1516-1558.  doi: 10.1515/anona-2020-0065.  Google Scholar

[28]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[29]

H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2017. Google Scholar

[30]

N. H. TuanV. V. Au and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Commun. Pure Appl. Anal., 20 (2021), 583-621.  doi: 10.3934/cpaa.2020282.  Google Scholar

[31]

N. H. Tuan and T. Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations, Proc. Amer. Math. Soc., 149 (2021), 143-161.  doi: 10.1090/proc/15131.  Google Scholar

[32]

N. H. Tuan and Y. Zhou, Well-posedness of an initial value problem for fractional diffusion equation with Caputo-Fabrizio derivative, J. Comput. Appl. Math., 375 (2020), 112811, 21 pp. doi: 10.1016/j.cam.2020.112811.  Google Scholar

[33]

R. XuT. ChenC. Liu and Y. Ding, Global well-posedness and global attractor of fourth order semilinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 1515-1529.  doi: 10.1002/mma.3165.  Google Scholar

[34]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Applied Numerical Mathematics, 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[35]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.  Google Scholar

[37]

H. YeQ. Liu and Z.-M. Chen, Global existence of solutions of the time fractional Cahn–Hilliard equation in $\mathbb R^3$, J. Evol. Equ., 21 (2021), 2377-2411.  doi: 10.1007/s00028-021-00687-1.  Google Scholar

[38]

X. Zheng, H. Wang and H. Fu, Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative, Chaos Solitons Fractals, 138 (2020), 109966, 7 pp. doi: 10.1016/j.chaos.2020.109966.  Google Scholar

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