February  2019, 39(2): 771-801. doi: 10.3934/dcds.2019032

Determination of initial data for a reaction-diffusion system with variable coefficients

1. 

Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh 700000, Vietnam

2. 

LaSIE, Facult des Sciences etTechnologies, Universi de La Rochelle, Avenue M. Crpeau, La Rochelle, Cedex17042, France

3. 

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author

Received  November 2017 Revised  July 2018 Published  November 2018

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in $L_2$ and $\stackrel{0}{H_1}$ norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

Citation: Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032
References:
[1]

W. BarthelC. John and F. Tröltzsch, Optimal boundary control of a system of reaction diffusion equations, ZAMM Z. Angew. Math. Mech., 90 (2010), 966-982. doi: 10.1002/zamm.200900359. Google Scholar

[2]

D. Bothe and G. Rolland, Global existence for a class of reaction-diffusion systems with mass action kinetics and concentration-dependent diffusivities, Acta Appl. Math., 139 (2015), 25-57. doi: 10.1007/s10440-014-9968-y. Google Scholar

[3]

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

[4]

L. C. Evans, Partial differential equations, Providence, Rhode Island: American Mathematical Society, 19 (1997), 451 pages.Google Scholar

[5]

S. Hapuarachchi and Y. Xu, Backward heat equation with time dependent variable coefficient, Mathematical Method in Applied Science, 40 (2016), 928-938. doi: 10.1002/mma.4022. Google Scholar

[6]

X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar

[7]

C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects for nonstandard linear diffusion problems, IMA J. Appl. Math., 70 (2005), 119-128. doi: 10.1093/imamat/hxh053. Google Scholar

[8]

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. Google Scholar

[9]

M. Kirane and S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations, Nonlinear Anal., 26 (1996), 1387-1396. doi: 10.1016/0362-546X(94)00337-H. Google Scholar

[10]

A. G. KlaasenA. Gene and C. T. William, Stationary wave solutions of a system of reactiondiffusion equations derived from the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 44 (1984), 96-110. doi: 10.1137/0144008. Google Scholar

[11]

Y. K. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competitiondiffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar

[12]

R. Lattès and J. L. Lions, Méthode de Quasi-réversibilité et Applications, Paris: Dunod, 1967. Google Scholar

[13]

P. T. Nam, An approximate solution for nonlinear backward parabolic equations, J. Math. Anal. Appl., 367 (2010), 337-349. doi: 10.1016/j.jmaa.2010.01.020. Google Scholar

[14]

G. Peano, Démonstration de l'intégrabilité des équations differentielles ordinaires, Math. Ann., 37 (1890), 182-228. doi: 10.1007/BF01200235. Google Scholar

[15]

B. Pena and C. Perez-Garcia, Stability of turing patterns in the brusselator model, Phs. Review E, 64 (2001), 056213, 9 pages. doi: 10.1103/PhysRevE.64.056213. Google Scholar

[16]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4. Google Scholar

[17]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1665-1700. Google Scholar

[18]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pages. doi: 10.1088/0266-5611/28/7/075007. Google Scholar

[19]

P. W. Schaefer, Energy bounds in nonstandard problems for parabolic systems, Nonlinear Anal., 63 (2005), 799-804. doi: 10.1016/j.na.2004.12.016. Google Scholar

[20]

D. D. Trong and N. H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal., 71 (2009), 4167-4176. doi: 10.1016/j.na.2009.02.092. Google Scholar

[21]

P. Zhou, On a Lotka-Volterra competition system: Diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp. doi: 10.1007/s00526-016-1082-8. Google Scholar

show all references

References:
[1]

W. BarthelC. John and F. Tröltzsch, Optimal boundary control of a system of reaction diffusion equations, ZAMM Z. Angew. Math. Mech., 90 (2010), 966-982. doi: 10.1002/zamm.200900359. Google Scholar

[2]

D. Bothe and G. Rolland, Global existence for a class of reaction-diffusion systems with mass action kinetics and concentration-dependent diffusivities, Acta Appl. Math., 139 (2015), 25-57. doi: 10.1007/s10440-014-9968-y. Google Scholar

[3]

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

[4]

L. C. Evans, Partial differential equations, Providence, Rhode Island: American Mathematical Society, 19 (1997), 451 pages.Google Scholar

[5]

S. Hapuarachchi and Y. Xu, Backward heat equation with time dependent variable coefficient, Mathematical Method in Applied Science, 40 (2016), 928-938. doi: 10.1002/mma.4022. Google Scholar

[6]

X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar

[7]

C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects for nonstandard linear diffusion problems, IMA J. Appl. Math., 70 (2005), 119-128. doi: 10.1093/imamat/hxh053. Google Scholar

[8]

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. Google Scholar

[9]

M. Kirane and S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations, Nonlinear Anal., 26 (1996), 1387-1396. doi: 10.1016/0362-546X(94)00337-H. Google Scholar

[10]

A. G. KlaasenA. Gene and C. T. William, Stationary wave solutions of a system of reactiondiffusion equations derived from the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 44 (1984), 96-110. doi: 10.1137/0144008. Google Scholar

[11]

Y. K. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competitiondiffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar

[12]

R. Lattès and J. L. Lions, Méthode de Quasi-réversibilité et Applications, Paris: Dunod, 1967. Google Scholar

[13]

P. T. Nam, An approximate solution for nonlinear backward parabolic equations, J. Math. Anal. Appl., 367 (2010), 337-349. doi: 10.1016/j.jmaa.2010.01.020. Google Scholar

[14]

G. Peano, Démonstration de l'intégrabilité des équations differentielles ordinaires, Math. Ann., 37 (1890), 182-228. doi: 10.1007/BF01200235. Google Scholar

[15]

B. Pena and C. Perez-Garcia, Stability of turing patterns in the brusselator model, Phs. Review E, 64 (2001), 056213, 9 pages. doi: 10.1103/PhysRevE.64.056213. Google Scholar

[16]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4. Google Scholar

[17]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1665-1700. Google Scholar

[18]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pages. doi: 10.1088/0266-5611/28/7/075007. Google Scholar

[19]

P. W. Schaefer, Energy bounds in nonstandard problems for parabolic systems, Nonlinear Anal., 63 (2005), 799-804. doi: 10.1016/j.na.2004.12.016. Google Scholar

[20]

D. D. Trong and N. H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal., 71 (2009), 4167-4176. doi: 10.1016/j.na.2009.02.092. Google Scholar

[21]

P. Zhou, On a Lotka-Volterra competition system: Diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp. doi: 10.1007/s00526-016-1082-8. Google Scholar

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