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April  2022, 27(4): 2345-2365. doi: 10.3934/dcdsb.2021135

On a generalized diffusion problem: A complex network approach

Laboratoire de Mathématiques Appliquées du Havre, Université du Havre Normandie, FR CNRS 3335, Institut des Systèmes Complexes de Normandie, 25, rue Philippe Lebon, BP 1123, 76063 Le Havre, Normandie, France

* Corresponding author: Guillaume Cantin

Received  February 2020 Revised  April 2021 Published  April 2022 Early access  May 2021

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

Citation: Guillaume Cantin, Alexandre Thorel. On a generalized diffusion problem: A complex network approach. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2345-2365. doi: 10.3934/dcdsb.2021135
References:
[1]

B. AmbrosioM. Aziz-Alaoui and V. Phan, Large time behaviour and synchronization of complex networks of reaction–diffusion systems of Fitzhugh–Nagumo type, IMA Journal of Applied Mathematics, 84 (2019), 416-443.  doi: 10.1093/imamat/hxy064.

[2]

M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226.  doi: 10.1016/B0-12-512666-2/00105-X.

[3]

I. BelykhM. HaslerM. Lauret and H. Nijmeijer, Synchronization and graph topology, International Journal of Bifurcation and Chaos, 15 (2005), 3423-3433.  doi: 10.1142/S0218127405014143.

[4]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, volume 151., Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.

[5]

B. Bialecki, A fourth order finite difference method for the Dirichlet biharmonic problem, Numerical Algorithms, 61 (2012), 351-375.  doi: 10.1007/s11075-012-9536-3.

[6]

G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.

[7]

G. CantinN. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA Journal of Applied Mathematics, 84 (2019), 974-1000.  doi: 10.1093/imamat/hxz022.

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[9]

G. ChenZ. Li and P. Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Advances in Computational Mathematics, 29 (2008), 113-133.  doi: 10.1007/s10444-007-9043-6.

[10]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, DCDS-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.

[11]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in a population, Journal of Mathematical Biology, 12 (1981), 237-249.  doi: 10.1007/BF00276132.

[12]

G. Da Prato, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Apple.(9), 54 (1975), 305-387. 

[13]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Università di Padova, 34 (1964), 205–223.

[14]

V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods, Society for Industrial and Applied Mathematics, 2015. doi: 10.1137/1.9781611974065.ch1.

[15]

G. Dore, $L^p$ regularity for abstract differential equations, In H. Komatsu, editor, Functional Analysis and Related Topics, 1991, pages 25–38, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg. doi: 10.1007/BFb0085472.

[16]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[17]

M. Ebenbeck and H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM Journal on Mathematical Analysis, 51 (2019), 1868-1912.  doi: 10.1137/18M1228104.

[18]

L. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM Journal on Numerical Analysis, 8 (1971), 278-287.  doi: 10.1137/0708029.

[19]

A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM Journal on Mathematical Analysis, 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[20]

R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional stokes problem, SIAM Review, 21 (1979), 167-212.  doi: 10.1137/1021028.

[21]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the american mathematical society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.

[22]

P. Grisvard, Équations différentielles abstraites, Annales scientifiques de l'Ecole Normale Superieure, 2 (1969), 311-395. 

[23]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 5 (1972), 657-729. 

[24]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.

[25]

D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, In Free Boundary Problems, Springer, 2003,189–196.

[26]

N. Katzourakis and T. Pryer, On the numerical approximation of $p$-biharmonic and $\infty$-biharmonic functions, Numerical Methods for Partial Differential Equations, 35 (2019), 155-180.  doi: 10.1002/num.22295.

[27]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.

[28]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete & Continuous Dynamical Systems - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.

[29]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.

[30]

R. Labbas and M. Moussaoui, On the resolution of the heat equation with discontinuous coefficients, Semigroup Forum, 60 (2000), 187-201.  doi: 10.1007/s002339910013.

[31]

F. Lara Ochoa, A generalized reaction diffusion model for spatial structure formed by motile cells, Biosystems, 17 (1984), 35-50.  doi: 10.1016/0303-2647(84)90014-5.

[32]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on S1, In Nonlinear Diffusion Equations and Their Equilibrium States II, 139–162, Math. Sci. Res. Inst. Publ., 13, Springer, New York, 1988. doi: 10.1007/978-1-4613-9608-6_8.

[33]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.

[34]

J. Morgan, Global existence for semilinear parabolic systems, SIAM journal on mathematical analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[35]

A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications Mathématiques de l'I.H.É.S., 19 (1964), 5–68.

[36]

A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis: Theory, Methods & Applications, 15 (1990), 797-814.  doi: 10.1016/0362-546X(90)90094-W.

[37]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980.

[38]

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

[39]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.

[40]

B. Rink and J. Sanders, Coupled cell networks: Semigroups, Lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509-3548.  doi: 10.1090/S0002-9947-2014-06221-1.

[41]

W. Ruan, Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity, Nonlinear Analysis, 182 (2019), 75-96.  doi: 10.1016/j.na.2018.12.003.

[42]

J. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Lecture Notes in Math., 1221 (1985), 195-222.  doi: 10.1007/BFb0099115.

[43]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258., Springer Science & Business Media, 1994. doi: 10.1007/978-1-4612-0873-0.

[44]

J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, Journal of Computational Physics, 55 (1984), 65-80.  doi: 10.1016/0021-9991(84)90015-9.

[45]

J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.

[46]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, Journal of Evolution Equations, 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.

[47]

A. Toselli and O. Widlund, Domain Decomposition Methods–Algorithms and Theory, Springer Series in Computational Mathematics, 34. Springer-Verlag, Berlin, 2005. doi: 10.1007/b137868.

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[49]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

[50]

J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection–diffusion equation, Communications in Numerical Methods in Engineering, 14 (1998), 209-218.  doi: 10.1002/(SICI)1099-0887(199803)14:3<209::AID-CNM139>3.0.CO;2-P.

show all references

References:
[1]

B. AmbrosioM. Aziz-Alaoui and V. Phan, Large time behaviour and synchronization of complex networks of reaction–diffusion systems of Fitzhugh–Nagumo type, IMA Journal of Applied Mathematics, 84 (2019), 416-443.  doi: 10.1093/imamat/hxy064.

[2]

M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226.  doi: 10.1016/B0-12-512666-2/00105-X.

[3]

I. BelykhM. HaslerM. Lauret and H. Nijmeijer, Synchronization and graph topology, International Journal of Bifurcation and Chaos, 15 (2005), 3423-3433.  doi: 10.1142/S0218127405014143.

[4]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, volume 151., Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.

[5]

B. Bialecki, A fourth order finite difference method for the Dirichlet biharmonic problem, Numerical Algorithms, 61 (2012), 351-375.  doi: 10.1007/s11075-012-9536-3.

[6]

G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.

[7]

G. CantinN. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA Journal of Applied Mathematics, 84 (2019), 974-1000.  doi: 10.1093/imamat/hxz022.

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[9]

G. ChenZ. Li and P. Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Advances in Computational Mathematics, 29 (2008), 113-133.  doi: 10.1007/s10444-007-9043-6.

[10]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, DCDS-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.

[11]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in a population, Journal of Mathematical Biology, 12 (1981), 237-249.  doi: 10.1007/BF00276132.

[12]

G. Da Prato, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Apple.(9), 54 (1975), 305-387. 

[13]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Università di Padova, 34 (1964), 205–223.

[14]

V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods, Society for Industrial and Applied Mathematics, 2015. doi: 10.1137/1.9781611974065.ch1.

[15]

G. Dore, $L^p$ regularity for abstract differential equations, In H. Komatsu, editor, Functional Analysis and Related Topics, 1991, pages 25–38, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg. doi: 10.1007/BFb0085472.

[16]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[17]

M. Ebenbeck and H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM Journal on Mathematical Analysis, 51 (2019), 1868-1912.  doi: 10.1137/18M1228104.

[18]

L. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM Journal on Numerical Analysis, 8 (1971), 278-287.  doi: 10.1137/0708029.

[19]

A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM Journal on Mathematical Analysis, 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[20]

R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional stokes problem, SIAM Review, 21 (1979), 167-212.  doi: 10.1137/1021028.

[21]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the american mathematical society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.

[22]

P. Grisvard, Équations différentielles abstraites, Annales scientifiques de l'Ecole Normale Superieure, 2 (1969), 311-395. 

[23]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 5 (1972), 657-729. 

[24]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.

[25]

D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, In Free Boundary Problems, Springer, 2003,189–196.

[26]

N. Katzourakis and T. Pryer, On the numerical approximation of $p$-biharmonic and $\infty$-biharmonic functions, Numerical Methods for Partial Differential Equations, 35 (2019), 155-180.  doi: 10.1002/num.22295.

[27]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.

[28]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete & Continuous Dynamical Systems - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.

[29]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.

[30]

R. Labbas and M. Moussaoui, On the resolution of the heat equation with discontinuous coefficients, Semigroup Forum, 60 (2000), 187-201.  doi: 10.1007/s002339910013.

[31]

F. Lara Ochoa, A generalized reaction diffusion model for spatial structure formed by motile cells, Biosystems, 17 (1984), 35-50.  doi: 10.1016/0303-2647(84)90014-5.

[32]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on S1, In Nonlinear Diffusion Equations and Their Equilibrium States II, 139–162, Math. Sci. Res. Inst. Publ., 13, Springer, New York, 1988. doi: 10.1007/978-1-4613-9608-6_8.

[33]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.

[34]

J. Morgan, Global existence for semilinear parabolic systems, SIAM journal on mathematical analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[35]

A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications Mathématiques de l'I.H.É.S., 19 (1964), 5–68.

[36]

A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis: Theory, Methods & Applications, 15 (1990), 797-814.  doi: 10.1016/0362-546X(90)90094-W.

[37]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980.

[38]

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

[39]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.

[40]

B. Rink and J. Sanders, Coupled cell networks: Semigroups, Lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509-3548.  doi: 10.1090/S0002-9947-2014-06221-1.

[41]

W. Ruan, Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity, Nonlinear Analysis, 182 (2019), 75-96.  doi: 10.1016/j.na.2018.12.003.

[42]

J. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Lecture Notes in Math., 1221 (1985), 195-222.  doi: 10.1007/BFb0099115.

[43]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258., Springer Science & Business Media, 1994. doi: 10.1007/978-1-4612-0873-0.

[44]

J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, Journal of Computational Physics, 55 (1984), 65-80.  doi: 10.1016/0021-9991(84)90015-9.

[45]

J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.

[46]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, Journal of Evolution Equations, 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.

[47]

A. Toselli and O. Widlund, Domain Decomposition Methods–Algorithms and Theory, Springer Series in Computational Mathematics, 34. Springer-Verlag, Berlin, 2005. doi: 10.1007/b137868.

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[49]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

[50]

J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection–diffusion equation, Communications in Numerical Methods in Engineering, 14 (1998), 209-218.  doi: 10.1002/(SICI)1099-0887(199803)14:3<209::AID-CNM139>3.0.CO;2-P.

Figure 1.  Splitting of a rectangular domain $ \Omega $ into a grid of sub-domains $ \omega_1, \, \dots, \, \omega_n $ in the dimension case $ N = 2 $
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Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087

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