February  2016, 9(1): 343-362. doi: 10.3934/dcdss.2016.9.343

Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis

1. 

Uni-CV, Cabo Verde and CEMAT, IST, Universidade de Lisboa, 1049-001 Lisbon, Portugal

2. 

Department of Mathematics and CEMAT/IST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisboa

3. 

Department of Mathematics and CEMAT/IST, Faculty of Sciences and Technology, University of Algarve, Campus de Gambelas 8005-139 Faro, Portugal

4. 

Dept Math and CEMAT, IST, Universidade de Lisboa, 1049-001 Lisbon, Portugal

Received  September 2014 Revised  February 2015 Published  December 2015

We study an atherosclerosis model described by a reaction-diffusion system of three equations, in one dimension, with homogeneous Neumann boundary conditions. The method of upper and lower solutions and its associated monotone iteration (the monotone iterative method) are used to establish existence, uniqueness and boundedness of global solutions for the problem. Upper and lower solutions are derived for the corresponding steady-state problem. Moreover, solutions of Cauchy problems defined for time-dependent system are presented as alternatives upper and lower solutions. The stability of constant steady-state solutions and the asymptotic behavior of the time-dependent solutions are studied.
Citation: Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
References:
[1]

N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press Inc., London, 1986.

[2]

V. Calvez, A. Ebde, N. Meunier and A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation, ESAIM Proceedings, 28 (2009), 1-12. doi: 10.1051/proc/2009036.

[3]

V. Calvez, J. Houot, N. Meunier, A. Raoult and G. Rusnakova, Mathematical and numerical modeling of early atherosclerotic lesions, ESAIM Proceedings, 30 (2010), 1-14. doi: 10.1051/proc/2010002.

[4]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin Heidelberg, 1979.

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, R.E. Krieger Pub. Co., 1983.

[6]

H. Daniel, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Heidelberg, 1981.

[7]

N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi and O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery, Elsevier, Computers & Fluids, 88 (2013), 826-833. doi: 10.1016/j.compfluid.2013.07.006.

[8]

W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model, PLoS ONE, 9 (2014), e90497. doi: 10.1371/journal.pone.0090497.

[9]

N. El Khatib, S. Genieys and V. Volpert, Atherosclerosis initiation modeled as an inflammatory process, Math. Model Nat. Phenom., 2 (2007), 126-141. doi: 10.1051/mmnp:2008022.

[10]

N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Mathematical modeling of atherosclerosis as an inflammatory disease, Phil. Trans. R. Soc. A, 367 (2009), 4877-4886. doi: 10.1098/rsta.2009.0142.

[11]

N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development, J. Math. Biol., 65 (2012), 349-374. doi: 10.1007/s00285-011-0461-1.

[12]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Math. Society, 1996. doi: 10.1090/gsm/012.

[13]

O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, American Math. Soc., 1968.

[14]

B. Liu and D. Tang, Computer simulations of atherosclerosis plaque growth in coronary arteries, Mol. Cell. Biomech., 7 (2010), 193-202.

[15]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[16]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Elsevier Science Ltd, Nonlinear Analysis, 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[17]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[18]

R. Ross, Atherosclerosis - an inflammatory disease, Massachussets Medical Soc., 340 (1999), 115-126.

[19]

F. Rothe, Global Solutions of Reaction-Diffusion System, Springer-Verlag, Berlin Heidelberg, 1984.

[20]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic bounded value problems, Indiana University Math. Journal, 21 (1972), 979-1000.

[21]

T. Silva, A. Sequeira, R. Santos and J. Tiago, Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow, Hindawi Publishing Corporation Conf. Papers in Math., 2013 (2013), Article ID 405914, 14 pages. doi: 10.1155/2013/405914.

show all references

References:
[1]

N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press Inc., London, 1986.

[2]

V. Calvez, A. Ebde, N. Meunier and A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation, ESAIM Proceedings, 28 (2009), 1-12. doi: 10.1051/proc/2009036.

[3]

V. Calvez, J. Houot, N. Meunier, A. Raoult and G. Rusnakova, Mathematical and numerical modeling of early atherosclerotic lesions, ESAIM Proceedings, 30 (2010), 1-14. doi: 10.1051/proc/2010002.

[4]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin Heidelberg, 1979.

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, R.E. Krieger Pub. Co., 1983.

[6]

H. Daniel, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Heidelberg, 1981.

[7]

N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi and O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery, Elsevier, Computers & Fluids, 88 (2013), 826-833. doi: 10.1016/j.compfluid.2013.07.006.

[8]

W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model, PLoS ONE, 9 (2014), e90497. doi: 10.1371/journal.pone.0090497.

[9]

N. El Khatib, S. Genieys and V. Volpert, Atherosclerosis initiation modeled as an inflammatory process, Math. Model Nat. Phenom., 2 (2007), 126-141. doi: 10.1051/mmnp:2008022.

[10]

N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Mathematical modeling of atherosclerosis as an inflammatory disease, Phil. Trans. R. Soc. A, 367 (2009), 4877-4886. doi: 10.1098/rsta.2009.0142.

[11]

N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development, J. Math. Biol., 65 (2012), 349-374. doi: 10.1007/s00285-011-0461-1.

[12]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Math. Society, 1996. doi: 10.1090/gsm/012.

[13]

O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, American Math. Soc., 1968.

[14]

B. Liu and D. Tang, Computer simulations of atherosclerosis plaque growth in coronary arteries, Mol. Cell. Biomech., 7 (2010), 193-202.

[15]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[16]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Elsevier Science Ltd, Nonlinear Analysis, 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[17]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[18]

R. Ross, Atherosclerosis - an inflammatory disease, Massachussets Medical Soc., 340 (1999), 115-126.

[19]

F. Rothe, Global Solutions of Reaction-Diffusion System, Springer-Verlag, Berlin Heidelberg, 1984.

[20]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic bounded value problems, Indiana University Math. Journal, 21 (1972), 979-1000.

[21]

T. Silva, A. Sequeira, R. Santos and J. Tiago, Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow, Hindawi Publishing Corporation Conf. Papers in Math., 2013 (2013), Article ID 405914, 14 pages. doi: 10.1155/2013/405914.

[1]

Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026

[2]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001

[3]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[4]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[5]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681

[6]

Pengchao Lai, Qi Li. Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3313-3323. doi: 10.3934/dcdsb.2021186

[7]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

[8]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[9]

Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25

[10]

Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209

[11]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114

[12]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[13]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control and Related Fields, 2022, 12 (1) : 147-168. doi: 10.3934/mcrf.2021005

[14]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[15]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[16]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[17]

Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122

[18]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

[19]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101

[20]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic and Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (220)
  • HTML views (1)
  • Cited by (2)

[Back to Top]