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doi: 10.3934/dcdsb.2020354

On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam, Department of Mathematics and Computer Science, University of Science Ho Chi Minh City-VNU, Vietnam

* Corresponding author: Nguyen Huy Tuan

Dedicated to Tomás Caraballo on his 60th birthday.

Received  June 2020 Revised  October 2020 Published  December 2020

We study for nonlinear Kirchhoff's model of pseudo parabolic type by considering its two different problems.

$ \bullet $ For initial value problem, we obtain the results on the existence and regularity of solutions. Moreover, we also prove that the solutions $ u $ corresponding with $ \beta < 1 $ of the problem convergence to $ u $ for $ \beta = 1 $.

$ \bullet $ For final value problem, we show that the ill-posed property in the sense of Hadamard is occurring. Using the Fourier truncation method to regularize the problem. We establish some stability estimates in the $ H^1 $ and $ L^p $ norms under some a-priori conditions on the sought solution.

Citation: Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020354
References:
[1]

R. M. P. AlmeidaS. N. Antontsev and J. C. M. Duque, On a nonlocal degenerate parabolic problem, Nonlinear Anal. RWA, 27 (2016), 146-157.  doi: 10.1016/j.nonrwa.2015.07.015.  Google Scholar

[2]

V. V. Au, M. Kirane and N. H. Tuan, On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms, Discrete Contin. Dyn. Syst. Ser. B, 174 (2020), 27 pages. Google Scholar

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G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965.  doi: 10.1016/j.na.2010.05.024.  Google Scholar

[4]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

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C. CaoM. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating $2$-D sphere: Gevrey regularity and asymptotic degrees of freedom, Z. Angew. Math. Phys., 50 (1999), 341-360.  doi: 10.1007/PL00001493.  Google Scholar

[6]

T. CaraballoH. CrauelJ. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.  Google Scholar

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T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Global attractor for a nonlocal $p$-Laplacian equation without uniqueness of solution, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1801-1816.  doi: 10.3934/dcdsb.2017107.  Google Scholar

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T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.  doi: 10.1016/j.na.2014.07.011.  Google Scholar

[9]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dynam., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.  Google Scholar

[10]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Global attractor for a nonlocal $p$-Laplacian equation without uniqueness of solution, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1801-1816.  doi: 10.3934/dcdsb.2017107.  Google Scholar

[11]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Asymptotic behaviour of nonlocal $p$-Laplacian reaction-diffusion problems, J. Math. Anal. Appl., 459 (2018), 997-1015.  doi: 10.1016/j.jmaa.2017.11.013.  Google Scholar

[12]

T. CaraballoJ. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Rev. Mat. Complut., 33 (2020), 583-617.  doi: 10.1007/s13163-019-00323-0.  Google Scholar

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A. S. CarassoJ. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal., 15 (1978), 344-367.  doi: 10.1137/0715023.  Google Scholar

[14]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.  Google Scholar

[15]

N.-H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 423-445.   Google Scholar

[16]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), Nonlinear Analysis: TMA, $\mathsf{30}$ (1997), 4619–4627. doi: 10.1016/S0362-546X(97)00169-7.  Google Scholar

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I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar

[18]

L. Dawidowski, The quasilinear parabolic Kirchhoff equation, Open Math., 15 (2017), 382-392.   Google Scholar

[19]

Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544.  doi: 10.1080/00036811.2015.1022153.  Google Scholar

[20]

M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates, J. Differential Equations, 245 (2008), 2979-3007.  doi: 10.1016/j.jde.2008.04.017.  Google Scholar

[21]

M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation, Math. Ann., 354 (2012), 1079-1102.  doi: 10.1007/s00208-011-0765-x.  Google Scholar

[22]

M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7.  Google Scholar

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[24]

S. KunduK. A. Pani and M. Khebchareon, On Kirchhoff's model of parabolic type, Numer. Funct. Anal. Optim., 37 (2016), 719-752.  doi: 10.1080/01630563.2016.1176930.  Google Scholar

[25]

Z. Liu and S. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl., 426 (2015), 267-287.  doi: 10.1016/j.jmaa.2015.01.044.  Google Scholar

[26]

L. A. MedeirosJ. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, I., J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.  Google Scholar

[27]

X. MingqiV. D. Rǎdulescu and B. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[28]

X. Peng, Y. Shang and X. Zheng, Pullback attractors of nonautonomous nonclassical diffusion equations with nonlocal diffusion, Z. Angew. Math. Phys., 69 (2018), Paper No. 110, 14 pp. doi: 10.1007/s00033-018-1005-y.  Google Scholar

[29]

C. A. RaposoM. SepúlvedaO. V. VillagránD. C. Pereira and M. L. Santos, Solution and asymptotic behaviour for a nonlocal coupled system of reaction-diffusion, Acta Appl. Math., 102 (2008), 37-56.  doi: 10.1007/s10440-008-9207-5.  Google Scholar

[30]

J. Simsen and J. Ferreira, A global attractor for a nonlocal parabolic problem, Nonlinear Stud., 21 (2014), 405-416.   Google Scholar

[31]

T. H. Skaggs and Z. J. Kabala, Recovering the history of a groundwater contaminant plume: Method of quasi-reversibility, Water Resources Research., 31 (1995), 2669-2673.   Google Scholar

[32]

N. H. TuanV. A. Khoa and V. A. Vo, Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM J. Math. Anal., 51 (2019), 60-85.  doi: 10.1137/18M1174064.  Google Scholar

[33]

N. H. TuanD. H. Q. Nam and T. M. N. Vo, On a backward problem for the Kirchhoff's model of parabolic type, Comput. Math. Appl., 77 (2019), 15-33.  doi: 10.1016/j.camwa.2018.08.072.  Google Scholar

[34]

N. H. Tuan, V. A. Vo, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems, 33 (2017), 055019, 40 pp. doi: 10.1088/1361-6420/aa635f.  Google Scholar

[35]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.   Google Scholar

show all references

References:
[1]

R. M. P. AlmeidaS. N. Antontsev and J. C. M. Duque, On a nonlocal degenerate parabolic problem, Nonlinear Anal. RWA, 27 (2016), 146-157.  doi: 10.1016/j.nonrwa.2015.07.015.  Google Scholar

[2]

V. V. Au, M. Kirane and N. H. Tuan, On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms, Discrete Contin. Dyn. Syst. Ser. B, 174 (2020), 27 pages. Google Scholar

[3]

G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965.  doi: 10.1016/j.na.2010.05.024.  Google Scholar

[4]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[5]

C. CaoM. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating $2$-D sphere: Gevrey regularity and asymptotic degrees of freedom, Z. Angew. Math. Phys., 50 (1999), 341-360.  doi: 10.1007/PL00001493.  Google Scholar

[6]

T. CaraballoH. CrauelJ. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.  Google Scholar

[7]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Global attractor for a nonlocal $p$-Laplacian equation without uniqueness of solution, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1801-1816.  doi: 10.3934/dcdsb.2017107.  Google Scholar

[8]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.  doi: 10.1016/j.na.2014.07.011.  Google Scholar

[9]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dynam., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.  Google Scholar

[10]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Global attractor for a nonlocal $p$-Laplacian equation without uniqueness of solution, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1801-1816.  doi: 10.3934/dcdsb.2017107.  Google Scholar

[11]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Asymptotic behaviour of nonlocal $p$-Laplacian reaction-diffusion problems, J. Math. Anal. Appl., 459 (2018), 997-1015.  doi: 10.1016/j.jmaa.2017.11.013.  Google Scholar

[12]

T. CaraballoJ. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Rev. Mat. Complut., 33 (2020), 583-617.  doi: 10.1007/s13163-019-00323-0.  Google Scholar

[13]

A. S. CarassoJ. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal., 15 (1978), 344-367.  doi: 10.1137/0715023.  Google Scholar

[14]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.  Google Scholar

[15]

N.-H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 423-445.   Google Scholar

[16]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), Nonlinear Analysis: TMA, $\mathsf{30}$ (1997), 4619–4627. doi: 10.1016/S0362-546X(97)00169-7.  Google Scholar

[17]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar

[18]

L. Dawidowski, The quasilinear parabolic Kirchhoff equation, Open Math., 15 (2017), 382-392.   Google Scholar

[19]

Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544.  doi: 10.1080/00036811.2015.1022153.  Google Scholar

[20]

M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates, J. Differential Equations, 245 (2008), 2979-3007.  doi: 10.1016/j.jde.2008.04.017.  Google Scholar

[21]

M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation, Math. Ann., 354 (2012), 1079-1102.  doi: 10.1007/s00208-011-0765-x.  Google Scholar

[22]

M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7.  Google Scholar

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[24]

S. KunduK. A. Pani and M. Khebchareon, On Kirchhoff's model of parabolic type, Numer. Funct. Anal. Optim., 37 (2016), 719-752.  doi: 10.1080/01630563.2016.1176930.  Google Scholar

[25]

Z. Liu and S. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl., 426 (2015), 267-287.  doi: 10.1016/j.jmaa.2015.01.044.  Google Scholar

[26]

L. A. MedeirosJ. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, I., J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.  Google Scholar

[27]

X. MingqiV. D. Rǎdulescu and B. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[28]

X. Peng, Y. Shang and X. Zheng, Pullback attractors of nonautonomous nonclassical diffusion equations with nonlocal diffusion, Z. Angew. Math. Phys., 69 (2018), Paper No. 110, 14 pp. doi: 10.1007/s00033-018-1005-y.  Google Scholar

[29]

C. A. RaposoM. SepúlvedaO. V. VillagránD. C. Pereira and M. L. Santos, Solution and asymptotic behaviour for a nonlocal coupled system of reaction-diffusion, Acta Appl. Math., 102 (2008), 37-56.  doi: 10.1007/s10440-008-9207-5.  Google Scholar

[30]

J. Simsen and J. Ferreira, A global attractor for a nonlocal parabolic problem, Nonlinear Stud., 21 (2014), 405-416.   Google Scholar

[31]

T. H. Skaggs and Z. J. Kabala, Recovering the history of a groundwater contaminant plume: Method of quasi-reversibility, Water Resources Research., 31 (1995), 2669-2673.   Google Scholar

[32]

N. H. TuanV. A. Khoa and V. A. Vo, Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM J. Math. Anal., 51 (2019), 60-85.  doi: 10.1137/18M1174064.  Google Scholar

[33]

N. H. TuanD. H. Q. Nam and T. M. N. Vo, On a backward problem for the Kirchhoff's model of parabolic type, Comput. Math. Appl., 77 (2019), 15-33.  doi: 10.1016/j.camwa.2018.08.072.  Google Scholar

[34]

N. H. Tuan, V. A. Vo, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems, 33 (2017), 055019, 40 pp. doi: 10.1088/1361-6420/aa635f.  Google Scholar

[35]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.   Google Scholar

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