American Institute of Mathematical Sciences

Advanced
June  2011, 4(3): 539-563. doi: 10.3934/dcdss.2011.4.539

Direct and inverse problems in age--structured population diffusion

Gabriella Di Blasio 1, and  Alfredo Lorenzi 2,

1. 

Dipartimento di Matematica, Università di Roma "La Sapienza", P.le A. Moro 5, 00185 Roma, Italy
2. 

Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano

Received  April 2009 Revised  November 2009 Published  November 2010

An identification problem for a class of ultraparabolic equations with a non local boundary condition, arising from age-dependent population diffusion, is analized. For such problems existence and uniqueness results as well as continuous dependence upon the data are proved. Regularity results with respect to space variables are also proved, using the theory of parabolic equations in $L^1$-spaces.
Keywords: population diffusion., ultraparabolic equations, Identification problems.
Mathematics Subject Classification: Primary: 35R30, 35K70; Secondary: 92D2.
Citation: Gabriella Di Blasio, Alfredo Lorenzi. Direct and inverse problems in age--structured population diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 539-563. doi: 10.3934/dcdss.2011.4.539
References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems,, Israel. J. Math., (1983), 225. Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura e Appl., IV (1996), 41. Google Scholar

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics,", Mathematical Modelling: Theory and Applications, 11 (2000). Google Scholar

[4]

A. Ashyralyev and P. E. Sobolevskii, "Well-Posedness of Parabolic Difference Equations,", Birkhäuser, (1994). Google Scholar

[5]

B. P. Ayati, A variable step method for an age-dependent population model with nonlinear diffusion,, SIAM J. Numer. Anal., (2000), 1571. Google Scholar

[6]

P. L. Butzer and H. Berens, "Semi-Groups of Operators and Approximation,", Springer-Verlag, (1967). Google Scholar

[7]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationelles,, J. Math. Pures et Appl., 54 (1975), 305. Google Scholar

[8]

G. Di Blasio, Linear parabolic equations in $L^p$-spaces,, Ann. Mat. Pura e Appl., IV (1984), 55. Google Scholar

[9]

G. Di Blasio, An ultraparabolic problem arising from age-dependent population diffusion,, Discrete Continuous Dynam. Systems - A, 25 (2009), 843. Google Scholar

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation,, Discrete Continuous Dynam. Systems - B, 7 (2007), 251. Google Scholar

[11]

J. Dyson, E. Sanchez, R. Villella-Bressan and G. F. Webb, An age and spatially structured model of tumor invasion with haptotaxis,, Discrete Continuous Dynam. Systems - B, 8 (2007), 45. Google Scholar

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics,, J. Evol. Equ., 2 (2002), 223. Google Scholar

[13]

A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$,, Discrete Continuous Dynam. Systems, 5 (1999), 663. Google Scholar

[14]

W. Rundell, Determining the death rate for an age-structured population from census data,, SIAM J. Appl. Math., 53 (1993), 1731. Google Scholar

[15]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators,", North-Holland, (1978). Google Scholar

[16]

G. F. Webb, Population models structured by age, size and position,, in, 1936 (2008), 1. Google Scholar

show all references

References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems,, Israel. J. Math., (1983), 225. Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura e Appl., IV (1996), 41. Google Scholar

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics,", Mathematical Modelling: Theory and Applications, 11 (2000). Google Scholar

[4]

A. Ashyralyev and P. E. Sobolevskii, "Well-Posedness of Parabolic Difference Equations,", Birkhäuser, (1994). Google Scholar

[5]

B. P. Ayati, A variable step method for an age-dependent population model with nonlinear diffusion,, SIAM J. Numer. Anal., (2000), 1571. Google Scholar

[6]

P. L. Butzer and H. Berens, "Semi-Groups of Operators and Approximation,", Springer-Verlag, (1967). Google Scholar

[7]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationelles,, J. Math. Pures et Appl., 54 (1975), 305. Google Scholar

[8]

G. Di Blasio, Linear parabolic equations in $L^p$-spaces,, Ann. Mat. Pura e Appl., IV (1984), 55. Google Scholar

[9]

G. Di Blasio, An ultraparabolic problem arising from age-dependent population diffusion,, Discrete Continuous Dynam. Systems - A, 25 (2009), 843. Google Scholar

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation,, Discrete Continuous Dynam. Systems - B, 7 (2007), 251. Google Scholar

[11]

J. Dyson, E. Sanchez, R. Villella-Bressan and G. F. Webb, An age and spatially structured model of tumor invasion with haptotaxis,, Discrete Continuous Dynam. Systems - B, 8 (2007), 45. Google Scholar

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics,, J. Evol. Equ., 2 (2002), 223. Google Scholar

[13]

A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$,, Discrete Continuous Dynam. Systems, 5 (1999), 663. Google Scholar

[14]

W. Rundell, Determining the death rate for an age-structured population from census data,, SIAM J. Appl. Math., 53 (1993), 1731. Google Scholar

[15]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators,", North-Holland, (1978). Google Scholar

[16]

G. F. Webb, Population models structured by age, size and position,, in, 1936 (2008), 1. Google Scholar

[1]

Gabriella Di Blasio. An ultraparabolic problem arising from age-dependent population diffusion. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 843-858. doi: 10.3934/dcds.2009.25.843
[2]

Gabriella Di Blasio. Ultraparabolic equations with nonlocal delayed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4945-4965. doi: 10.3934/dcds.2013.33.4945
[3]

Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
[4]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428
[5]

Hedia Fgaier, Hermann J. Eberl. Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation. Conference Publications, 2009, 2009 (Special) : 230-239. doi: 10.3934/proc.2009.2009.230
[6]

Said Boulite, S. Hadd, L. Maniar. Critical spectrum and stability for population equations with diffusion in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 265-276. doi: 10.3934/dcdsb.2005.5.265
[7]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[8]

Alfredo Lorenzi. Identification problems related to cylindrical dielectrics **in presence of polarization**. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2247-2265. doi: 10.3934/dcdsb.2014.19.2247
[9]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41
[10]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19
[11]

Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765
[12]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[13]

Hirotoshi Kuroda, Noriaki Yamazaki. Approximating problems of vectorial singular diffusion equations with inhomogeneous terms and numerical simulations. Conference Publications, 2009, 2009 (Special) : 486-495. doi: 10.3934/proc.2009.2009.486
[14]

Martin Burger, Jan-Frederik Pietschmann, Marie-Therese Wolfram. Identification of nonlinearities in transport-diffusion models of crowded motion. Inverse Problems & Imaging, 2013, 7 (4) : 1157-1182. doi: 10.3934/ipi.2013.7.1157
[15]

Gianluca Mola, Noboru Okazawa, Jan Prüss, Tomomi Yokota. Semigroup-theoretic approach to identification of linear diffusion coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 777-790. doi: 10.3934/dcdss.2016028
[16]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749
[17]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[18]

David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control & Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391
[19]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[20]

Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences