# American Institute of Mathematical Sciences

March  2021, 26(3): 1579-1613. doi: 10.3934/dcdsb.2020174

## On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms

 1 Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam 2 Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam 3 LaSIE, Faculté des Sciences, Pole Sciences et Technologies, Université de La Rochelle, Avenue M. Crepeau, 17042 La Rochelle Cedex, France, NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia, RUDN University, 6 Miklukho-Maklay St, Moscow 117198, Russia 4 Applied Analysis Research Group, Faculty of Mathematics, and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: nguyenhuytuan@tdtu.edu.vn

Received  December 2019 Published  March 2021 Early access  May 2020

We study a terminal value parabolic system with nonlinear-nonlocal diffusions. Firstly, we consider the issue of existence and ill-posed property of a solution. Then we introduce two regularization methods to solve the system in which the diffusion coefficients are globally Lipschitz or locally Lipschitz under some a priori assumptions on the sought solutions. The existence, uniqueness and regularity of solutions of the regularized problem are obtained. Furthermore, The error estimates show that the approximate solution converges to the exact solution in $L^2$ norm and also in $H^1$ norm.

Citation: 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
##### References:
 [1] R. M. P. Almeida, S. N. Antontsev, J. C. M. Duque and J. A. Ferreira, A reaction-diffusion model for the non-local coupled system: Existence, uniqueness, long-time behaviour and localization properties of solutions, IMA J. Appl. Math., 81 (2016), 344-364.  doi: 10.1093/imamat/hxv041. [2] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.  doi: 10.1016/j.camwa.2005.01.008. [3] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2. [4] G. Autuori, P. 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] G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl., 352 (2009), 149-165.  doi: 10.1016/j.jmaa.2008.04.066. [6] 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. [7] G. Avalos, I. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchhoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8. [8] S. Boulaaras and A. Allahem, Existence of positive solutions of nonlocal $p(x)$-Kirchhoff evolutionary systems via sub-super solutions concept, Symmetry, 11 (2019), 11pp. doi: 10.3390/sym11020253. [9] M. Camurdan and R. Triggiani, Sharp regularity of a coupled system of a wave and a Kirchhoff equation with point control arising in noise reduction, Differential Integral Equations, 12 (1999), 101-118. [10] T. Caraballo, M. Herrera-Cobos and P. Martín-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. [11] T. Caraballo, M. 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. [12] T. Caraballo, M. 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. [13] T. Caraballo, M. Herrera-Cobos and P. M. Rubio, Asymptotic behaviour of nonlocal $p$-Laplacian reactiondiffusion problems, J. Math. Anal. Appl., 459 (2018), 997-1015.  doi: 10.1016/j.jmaa.2017.11.013. [14] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627.  doi: 10.1016/S0362-546X(97)00169-7. [15] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [16] ç Dawidowski, The quasilinear parabolic Kirchhoff equation, Open Math., 15 (2017) 382–392. doi: 10.1515/math-2017-0036. [17] 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. [18] 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. [19] 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. [20] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Methods Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. [21] A. Hajej, Z. Hajjej and L. Tebou, Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping, J. Math. Anal. Appl., 474 (2019), 290-308.  doi: 10.1016/j.jmaa.2019.01.046. [22] E. J. Hurtado, O. H. Miyagaki and R. d. S. Rodrigues, Existence and asymptotic behaviour for a Kirchhoff type equation with variable critical growth exponent, Milan J. Math., 85 (2017), 71-102.  doi: 10.1007/s00032-017-0266-9. [23] J. I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems with a balance law and nonlinearities of exponential growth, J. Differential Equations, 165 (2000), 24-41.  doi: 10.1006/jdeq.2000.3769. [24] M. Kirane, Global bounds and asymptotics for a system of reaction-diffusion equations, J. Math. Anal. Appl., 138 (1989), 328-342.  doi: 10.1016/0022-247X(89)90293-X. [25] M. Kirane and M. Qafsaoui, Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 268 (2002), 217-243.  doi: 10.1006/jmaa.2001.7819. [26] J. Límaco, H. R. Clark and L. A. Medeiros, On damped Kirchhoff equation with variable coefficients, J. Math. Anal. Appl., 307 (2005), 641-655.  doi: 10.1016/j.jmaa.2004.12.032. [27] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.  doi: 10.1016/j.na.2005.03.021. [28] H. Medekhelz, S. Boulaaras and R. Guefaifia, Existence of positive solutions for a class of Kirchhoff parabolic systems with multiple parameters, Applied Math. E-Notes, 18 (2018), 295-306. [29] C. A. Raposo, M. Sepúlveda, O. V. Villagrán, D. 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] N. H. Tuan, V. V. Au, 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), 40pp. doi: 10.1088/1361-6420/aa635f. [32] H. T. Nguyen, V. A. Khoa and 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. [33] N. H. Tuan, D. 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.

show all references

##### References:
 [1] R. M. P. Almeida, S. N. Antontsev, J. C. M. Duque and J. A. Ferreira, A reaction-diffusion model for the non-local coupled system: Existence, uniqueness, long-time behaviour and localization properties of solutions, IMA J. Appl. Math., 81 (2016), 344-364.  doi: 10.1093/imamat/hxv041. [2] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.  doi: 10.1016/j.camwa.2005.01.008. [3] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2. [4] G. Autuori, P. 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] G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl., 352 (2009), 149-165.  doi: 10.1016/j.jmaa.2008.04.066. [6] 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. [7] G. Avalos, I. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchhoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8. [8] S. Boulaaras and A. Allahem, Existence of positive solutions of nonlocal $p(x)$-Kirchhoff evolutionary systems via sub-super solutions concept, Symmetry, 11 (2019), 11pp. doi: 10.3390/sym11020253. [9] M. Camurdan and R. Triggiani, Sharp regularity of a coupled system of a wave and a Kirchhoff equation with point control arising in noise reduction, Differential Integral Equations, 12 (1999), 101-118. [10] T. Caraballo, M. Herrera-Cobos and P. Martín-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. [11] T. Caraballo, M. 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. [12] T. Caraballo, M. 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. [13] T. Caraballo, M. Herrera-Cobos and P. M. Rubio, Asymptotic behaviour of nonlocal $p$-Laplacian reactiondiffusion problems, J. Math. Anal. Appl., 459 (2018), 997-1015.  doi: 10.1016/j.jmaa.2017.11.013. [14] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627.  doi: 10.1016/S0362-546X(97)00169-7. [15] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [16] ç Dawidowski, The quasilinear parabolic Kirchhoff equation, Open Math., 15 (2017) 382–392. doi: 10.1515/math-2017-0036. [17] 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. [18] 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. [19] 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. [20] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Methods Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. [21] A. Hajej, Z. Hajjej and L. Tebou, Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping, J. Math. Anal. Appl., 474 (2019), 290-308.  doi: 10.1016/j.jmaa.2019.01.046. [22] E. J. Hurtado, O. H. Miyagaki and R. d. S. Rodrigues, Existence and asymptotic behaviour for a Kirchhoff type equation with variable critical growth exponent, Milan J. Math., 85 (2017), 71-102.  doi: 10.1007/s00032-017-0266-9. [23] J. I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems with a balance law and nonlinearities of exponential growth, J. Differential Equations, 165 (2000), 24-41.  doi: 10.1006/jdeq.2000.3769. [24] M. Kirane, Global bounds and asymptotics for a system of reaction-diffusion equations, J. Math. Anal. Appl., 138 (1989), 328-342.  doi: 10.1016/0022-247X(89)90293-X. [25] M. Kirane and M. Qafsaoui, Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 268 (2002), 217-243.  doi: 10.1006/jmaa.2001.7819. [26] J. Límaco, H. R. Clark and L. A. Medeiros, On damped Kirchhoff equation with variable coefficients, J. Math. Anal. Appl., 307 (2005), 641-655.  doi: 10.1016/j.jmaa.2004.12.032. [27] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.  doi: 10.1016/j.na.2005.03.021. [28] H. Medekhelz, S. Boulaaras and R. Guefaifia, Existence of positive solutions for a class of Kirchhoff parabolic systems with multiple parameters, Applied Math. E-Notes, 18 (2018), 295-306. [29] C. A. Raposo, M. Sepúlveda, O. V. Villagrán, D. 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] N. H. Tuan, V. V. Au, 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), 40pp. doi: 10.1088/1361-6420/aa635f. [32] H. T. Nguyen, V. A. Khoa and 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. [33] N. H. Tuan, D. 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.
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