June  2022, 42(6): 2637-2665. doi: 10.3934/dcds.2021206

Global well-posedness for fractional Sobolev-Galpern type equations

1. 

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

2. 

Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

3. 

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

* Corresponding author: yangchao_@hrbeu.edu.cn (Chao Yang)

Other authors: nguyenhuytuan@vlu.edu.vn (Huy Tuan Nguyen), nguyenanhtuan@vlu.edu.vn (Nguyen Anh Tuan).

Received  August 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.

Citation: Huy Tuan Nguyen, Nguyen Anh Tuan, Chao Yang. Global well-posedness for fractional Sobolev-Galpern type equations. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2637-2665. doi: 10.3934/dcds.2021206
References:
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J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 63 (1986), 117-134.  doi: 10.1016/0022-0396(86)90057-4.

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C. J. AmickJ. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.

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E. BazhlekovaB. JinR. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.

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T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

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M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

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T. CaraballoA. M. M. Duran and F. Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1817-1833.  doi: 10.3934/dcdsb.2017108.

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A. O. CelebiV. K. Kalantarov and M. Polat, Attractors for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 157 (1999), 439-451.  doi: 10.1006/jdeq.1999.3634.

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

[9]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

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P. J. Chen and E. G. Morton, On a theory of heat conduction involving two temperatures, Zeitschrift für Angewandte Mathematik und Physik, 19 (1968), 614-627.  doi: 10.1007/BF01594969.

[11]

Y. ChenH. GaoM. J. 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 Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

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B. D. Coleman and W. Noll, An approximation theorem for functionals, with applications in continuum mechanics, Arch. Rational Mech. Anal., 6 (1960), 355-370.  doi: 10.1007/BF00276168.

[13]

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.

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R. Grande, Space-time fractional nonlinear Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 4172-4212.  doi: 10.1137/19M1247140.

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N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differential Equations, 251 (2011), 1172-1194.  doi: 10.1016/j.jde.2011.02.015.

[16]

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

[17]

W. LianW. Juan and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

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L. A. Medeiros and G. P. Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal., 8 (1977), 792-799.  doi: 10.1137/0508062.

[19]

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

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I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[21]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.

[22]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[23]

N. Trudinger, On imbeddings into orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[24]

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.

[25]

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.

[26]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[27]

Y. Xiao, Packing measure of the sample paths of fractional Brownian motion, Trans. Amer. Math. Soc., 348 (1996), 3193-3213.  doi: 10.1090/S0002-9947-96-01712-6.

[28]

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.

[29]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

[30]

X. ZhuF. Li and T. Rong, Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.  doi: 10.3934/cpaa.2015.14.2465.

show all references

References:
[1]

J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 63 (1986), 117-134.  doi: 10.1016/0022-0396(86)90057-4.

[2]

C. J. AmickJ. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.

[3]

E. BazhlekovaB. JinR. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.

[4]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[5]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[6]

T. CaraballoA. M. M. Duran and F. Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1817-1833.  doi: 10.3934/dcdsb.2017108.

[7]

A. O. CelebiV. K. Kalantarov and M. Polat, Attractors for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 157 (1999), 439-451.  doi: 10.1006/jdeq.1999.3634.

[8]

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

[9]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[10]

P. J. Chen and E. G. Morton, On a theory of heat conduction involving two temperatures, Zeitschrift für Angewandte Mathematik und Physik, 19 (1968), 614-627.  doi: 10.1007/BF01594969.

[11]

Y. ChenH. GaoM. J. 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 Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

[12]

B. D. Coleman and W. Noll, An approximation theorem for functionals, with applications in continuum mechanics, Arch. Rational Mech. Anal., 6 (1960), 355-370.  doi: 10.1007/BF00276168.

[13]

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.

[14]

R. Grande, Space-time fractional nonlinear Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 4172-4212.  doi: 10.1137/19M1247140.

[15]

N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differential Equations, 251 (2011), 1172-1194.  doi: 10.1016/j.jde.2011.02.015.

[16]

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

[17]

W. LianW. Juan and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[18]

L. A. Medeiros and G. P. Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal., 8 (1977), 792-799.  doi: 10.1137/0508062.

[19]

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

[20]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[21]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.

[22]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[23]

N. Trudinger, On imbeddings into orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[24]

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.

[25]

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.

[26]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[27]

Y. Xiao, Packing measure of the sample paths of fractional Brownian motion, Trans. Amer. Math. Soc., 348 (1996), 3193-3213.  doi: 10.1090/S0002-9947-96-01712-6.

[28]

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.

[29]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

[30]

X. ZhuF. Li and T. Rong, Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.  doi: 10.3934/cpaa.2015.14.2465.

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