March  2022, 21(3): 817-835. doi: 10.3934/cpaa.2021200

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

* Corresponding author

Received  August 2021 Published  March 2022 Early access  December 2021

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.07. The first author was also supported by the Vietnam Institute for Advanced Study in Mathematics-VIASM

We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.

Citation: Dinh-Ke Tran, Nhu-Thang Nguyen. On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations. Communications on Pure and Applied Analysis, 2022, 21 (3) : 817-835. doi: 10.3934/cpaa.2021200
References:
[1]

A. Allaberen, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002.

[2]

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.

[3]

P. CannarsaH. Frankowska and E. M. Marchini, Optimal control for evolution equations with memory, J. Evol. Equ., 13 (2013), 197-227.  doi: 10.1007/s00028-013-0175-5.

[4]

J. R. Cannon and Y.P. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.  doi: 10.1137/0727036.

[5]

Ph. 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.

[6]

M. ContiElsa M. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986.  doi: 10.1090/S0002-9947-2013-06097-7.

[7]

G. Di Blasio, Parabolic Volterra integrodifferential equations of convolution type, J. Integral Equ. Appl., 6 (1994), 479-508.  doi: 10.1216/jiea/1181075833.

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007.

[9]

K. Ezzinbi, S. Ghnimi and M. A. Taoudi, Existence results for some nonlocal partial integrodifferential equations without compactness or equicontinuity, J. Fixed Point Theory Appl., 21 (2019), 24 pp. doi: 10.1007/s11784-019-0689-8.

[10]

L. C. Evans, Partial Differential Equations, Second edition. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[11]

T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces, J. Math. Anal. Appl., 483 (2020), 123655, 23 pp. doi: 10.1016/j.jmaa.2019.123655.

[12]

D. Lan, Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations, Evol. Equ. Control Theory, 2021. doi: 10.3934/eect.2021002.

[13]

N.H. Luc, D. Lan, D. O'Regan, N.A. Tuan and Y. Zhou, On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation, J. Fixed Point Theory Appl., 23 (2021), 28 pp. doi: 10.1007/s11784-021-00897-7.

[14]

N. H. LucN. H. Tuan and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.

[15]

S. McKee and A. Stokes, Product integration methods for the nonlinear Basset equation, SIAM J. Numer. Anal., 20 (1983), 143-160.  doi: 10.1137/0720010.

[16]

R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl., 22 (1968), 319-340.  doi: 10.1016/0022-247X(68)90176-5.

[17]

R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 313-332.  doi: 10.1016/0022-247X(78)90234-2.

[18]

A. Mohebbi, Crank-Nicolson and Legendre spectral collocation methods for a partial integro-differential equation with a singular kernel, J. Comput. Appl. Math., 349 (2019), 197-206.  doi: 10.1016/j.cam.2018.09.034.

[19]

T. B. NgocN. H. LucV. V. AuN. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., 44 (2020), 2532-2558.  doi: 10.1002/mma.6162.

[20]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[21]

N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 18 pp. doi: 10.1016/j.cnsns.2019.104873.

[22]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[23]

B. Wu and J. Yu, Uniqueness of an inverse problem for an integro-differential equation related to the Basset problem, Bound. Value Probl., 229 (2014), 9 pp. doi: 10.1186/s13661-014-0229-9.

[24]

J. ZierepR. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.

[25]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Methods Appl. Sci., 44 (2021), 2431-2438.  doi: 10.1002/mma.5926.

show all references

References:
[1]

A. Allaberen, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002.

[2]

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.

[3]

P. CannarsaH. Frankowska and E. M. Marchini, Optimal control for evolution equations with memory, J. Evol. Equ., 13 (2013), 197-227.  doi: 10.1007/s00028-013-0175-5.

[4]

J. R. Cannon and Y.P. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.  doi: 10.1137/0727036.

[5]

Ph. 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.

[6]

M. ContiElsa M. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986.  doi: 10.1090/S0002-9947-2013-06097-7.

[7]

G. Di Blasio, Parabolic Volterra integrodifferential equations of convolution type, J. Integral Equ. Appl., 6 (1994), 479-508.  doi: 10.1216/jiea/1181075833.

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007.

[9]

K. Ezzinbi, S. Ghnimi and M. A. Taoudi, Existence results for some nonlocal partial integrodifferential equations without compactness or equicontinuity, J. Fixed Point Theory Appl., 21 (2019), 24 pp. doi: 10.1007/s11784-019-0689-8.

[10]

L. C. Evans, Partial Differential Equations, Second edition. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[11]

T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces, J. Math. Anal. Appl., 483 (2020), 123655, 23 pp. doi: 10.1016/j.jmaa.2019.123655.

[12]

D. Lan, Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations, Evol. Equ. Control Theory, 2021. doi: 10.3934/eect.2021002.

[13]

N.H. Luc, D. Lan, D. O'Regan, N.A. Tuan and Y. Zhou, On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation, J. Fixed Point Theory Appl., 23 (2021), 28 pp. doi: 10.1007/s11784-021-00897-7.

[14]

N. H. LucN. H. Tuan and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.

[15]

S. McKee and A. Stokes, Product integration methods for the nonlinear Basset equation, SIAM J. Numer. Anal., 20 (1983), 143-160.  doi: 10.1137/0720010.

[16]

R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl., 22 (1968), 319-340.  doi: 10.1016/0022-247X(68)90176-5.

[17]

R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 313-332.  doi: 10.1016/0022-247X(78)90234-2.

[18]

A. Mohebbi, Crank-Nicolson and Legendre spectral collocation methods for a partial integro-differential equation with a singular kernel, J. Comput. Appl. Math., 349 (2019), 197-206.  doi: 10.1016/j.cam.2018.09.034.

[19]

T. B. NgocN. H. LucV. V. AuN. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., 44 (2020), 2532-2558.  doi: 10.1002/mma.6162.

[20]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[21]

N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 18 pp. doi: 10.1016/j.cnsns.2019.104873.

[22]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[23]

B. Wu and J. Yu, Uniqueness of an inverse problem for an integro-differential equation related to the Basset problem, Bound. Value Probl., 229 (2014), 9 pp. doi: 10.1186/s13661-014-0229-9.

[24]

J. ZierepR. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.

[25]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Methods Appl. Sci., 44 (2021), 2431-2438.  doi: 10.1002/mma.5926.

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