October  2021, 26(10): 5465-5494. 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  October 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (10) : 5465-5494. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[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.

[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.

[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.

[35]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[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.

[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.

[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.

[35]

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

[1]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[2]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005

[3]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005

[4]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7185-7206. doi: 10.3934/dcdsb.2022039

[5]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[6]

Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631

[7]

Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems and Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020

[8]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations and Control Theory, 2022, 11 (1) : 225-238. doi: 10.3934/eect.2020109

[9]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[10]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

[11]

William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial and Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291

[12]

Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092

[13]

Jong-Shenq Guo, Nikos I. Kavallaris. On a nonlocal parabolic problem arising in electrostatic MEMS control. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1723-1746. doi: 10.3934/dcds.2012.32.1723

[14]

Gernot Holler, Karl Kunisch. Learning nonlocal regularization operators. Mathematical Control and Related Fields, 2022, 12 (1) : 81-114. doi: 10.3934/mcrf.2021003

[15]

Qiang Du, Jiang Yang, Zhi Zhou. Analysis of a nonlocal-in-time parabolic equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016

[16]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315

[17]

Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443

[18]

Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218

[19]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[20]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (281)
  • HTML views (305)
  • Cited by (0)

Other articles
by authors

[Back to Top]