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February  2021, 20(2): 583-621. doi: 10.3934/cpaa.2020282

Semilinear Caputo time-fractional pseudo-parabolic equations

1. 

Department of Mathematics and Computer Science, University of Science Ho Chi Minh City, Vietnam

2. 

Vietnam National University, Ho Chi Minh City, Vietnam

3. 

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

4. 

Institute of Fundamental and Applied Sciences, Duy Tan University Ho Chi Minh City, 700000, Vietnam

5. 

Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

6. 

College of Mathematical Sciences, Harbin Engineering University, 150001, China

*Corresponding author

Received  July 2020 Revised  September 2020 Published  December 2020

Fund Project: The first and the second author were supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09. The third author was supported by National Natural Science Foundation of China (11871017)

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

Citation: Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
References:
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J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, T. Am. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.  Google Scholar

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G. Akagi, Fractional flows driven by subdifferentials in Hilbert spaces, Israel J. Math., 234 (2019), 809-862.  doi: 10.1007/s11856-019-1936-9.  Google Scholar

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B. de Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.  Google Scholar

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H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772.  Google Scholar

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J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587.   Google Scholar

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E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062.  Google Scholar

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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

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C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

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show all references

References:
[1]

J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, T. Am. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.  Google Scholar

[2] R. P. AgarwalM. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar
[3]

G. Akagi, Fractional flows driven by subdifferentials in Hilbert spaces, Israel J. Math., 234 (2019), 809-862.  doi: 10.1007/s11856-019-1936-9.  Google Scholar

[4]

B. de Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.  Google Scholar

[5]

B. Andrade and A. Viana, On a fractional reaction-diffusion equation, Z. Angew. Math. Phys., 68 (2017), 11 pp. doi: 10.1007/s00033-017-0801-0.  Google Scholar

[6]

S. Antontsev and S. Shmarev, On a class of fully nonlinear parabolic equations, Adv. Nonlinear Anal., 8 (2019), 79-100.  doi: 10.1515/anona-2016-0055.  Google Scholar

[7]

H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772.  Google Scholar

[8]

J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587.   Google Scholar

[9]

E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062.  Google Scholar

[10]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Geophys. J. Int., 13 (1967), 529-539.   Google Scholar

[11]

Y. CaoJ. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equ., 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar

[12]

Y. Cao and J. X. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 631-642.  doi: 10.3934/dcds.2016.36.631.  Google Scholar

[13]

D. del Castillo-Negrete, B. A. Carreras and V. E. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas, 11, 3854 (2004). Google Scholar

[14]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[15]

Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 39pp. doi: 10.1016/j.na.2019.111664.  Google Scholar

[16]

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

[17]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[18]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.  Google Scholar

[19]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.  Google Scholar

[20]

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

[21]

M. Fila and J. Lankeit, Lack of smoothing for bounded solutions of a semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 1437-1452.  doi: 10.1515/anona-2020-0059.  Google Scholar

[22]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[23]

T-E. GhoulN. V. Tien and H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 388-412.  doi: 10.1515/anona-2020-0006.  Google Scholar

[24]

V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972), 57-78.  doi: 10.1007/BF00281474.  Google Scholar

[25]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equ., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.  Google Scholar

[26]

R. GorenfloY. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414.   Google Scholar

[27]

R. Gorenflo, A. A. Kilbas and F. Mainardi, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

[28]

E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Second printing corrected, Springer-Verlag, Berlin, (1969).  Google Scholar

[29]

L. JinL. Li and S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221-2232.  doi: 10.1016/j.camwa.2017.03.005.  Google Scholar

[30]

V. Keyantuo and M. Warma, On the interior approximate controllability for fractional wave equations, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 3719-3739.  doi: 10.3934/dcds.2016.36.3719.  Google Scholar

[31]

S. Khomrutai, Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differ. Equ., 260 (2015), 3598-3657.  doi: 10.1016/j.jde.2015.10.043.  Google Scholar

[32]

L. LiJ. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differ. Equ., 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.  Google Scholar

[33]

Z. P. Li and W. J. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181-3203.  doi: 10.1007/s00033-015-0558-2.  Google Scholar

[34]

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

[35]

S. JiJ. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar

[36]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equ., 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar

[37]

A. MaganaA. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.  Google Scholar

[38]

B. B. Mandelbrot and J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar

[39]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[40]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal., 54 (2016), 848-873.  doi: 10.1137/14096308X.  Google Scholar

[41]

S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electronic Research Archive, 27 (2019), 89-99.  doi: 10.3934/era.2019011.  Google Scholar

[42]

N. PanP. PucciR. Xu and B. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evol. Equ., 19 (2019), 615-643.  doi: 10.1007/s00028-019-00489-6.  Google Scholar

[43]

N. S. PapageorgiouaeV. D. Rădulescu and D. D. Repovă, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl., 136 (2020), 1-21.  doi: 10.1016/j.matpur.2020.02.004.  Google Scholar

[44]

V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, T. Am. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[45]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 (1998), Elsevier, Amsterdam.  Google Scholar

[46]

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