American Institute of Mathematical Sciences

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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

Telma Silva 1, , Adélia Sequeira 2, , Rafael F. Santos 3, and  Jorge Tiago 4,

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.
Keywords: stability, asymptotic behavior of solutions., reaction-diffusion equations, quasimonotone property, Atherosclerosis, upper and lower solutions.
Mathematics Subject Classification: Primary: 35K57, 92C50; Secondary: 35A01, 35A08, 35B35, 35B4.
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.

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