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

Direct and inverse problems in age--structured population diffusion

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.
Citation: Gabriella Di Blasio, Alfredo Lorenzi. Direct and inverse problems in age--structured population diffusion. Discrete and 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., 45 (1983), 225-254.

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., IV (1996), 41-62.

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics," Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publisher, Dordrecht, 2000.

[4]

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

[5]

B. P. Ayati, A variable step method for an age-dependent population model with nonlinear diffusion, SIAM J. Numer. Anal., 37 (2000), 1571-1589.

[6]

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

[7]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationelles, J. Math. Pures et Appl., 54 (1975), 305-387.

[8]

G. Di Blasio, Linear parabolic equations in $L^p$-spaces, Ann. Mat. Pura e Appl., IV (1984), 55-104.

[9]

G. Di Blasio, An ultraparabolic problem arising from age-dependent population diffusion, Discrete Continuous Dynam. Systems - A, 25 (2009), 843-858.

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273.

[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), 45-60.

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics, J. Evol. Equ., 2 (2002), 223-239.

[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), 663-683.

[14]

W. Rundell, Determining the death rate for an age-structured population from census data, SIAM J. Appl. Math., 53 (1993), 1731-1746.

[15]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland, 1978.

[16]

G. F. Webb, Population models structured by age, size and position, in "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics, Vol. 1936, Springer-Verlag, Berlin-New York, (2008), 1-49.

show all references

References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel. J. Math., 45 (1983), 225-254.

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., IV (1996), 41-62.

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics," Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publisher, Dordrecht, 2000.

[4]

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

[5]

B. P. Ayati, A variable step method for an age-dependent population model with nonlinear diffusion, SIAM J. Numer. Anal., 37 (2000), 1571-1589.

[6]

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

[7]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationelles, J. Math. Pures et Appl., 54 (1975), 305-387.

[8]

G. Di Blasio, Linear parabolic equations in $L^p$-spaces, Ann. Mat. Pura e Appl., IV (1984), 55-104.

[9]

G. Di Blasio, An ultraparabolic problem arising from age-dependent population diffusion, Discrete Continuous Dynam. Systems - A, 25 (2009), 843-858.

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273.

[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), 45-60.

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics, J. Evol. Equ., 2 (2002), 223-239.

[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), 663-683.

[14]

W. Rundell, Determining the death rate for an age-structured population from census data, SIAM J. Appl. Math., 53 (1993), 1731-1746.

[15]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland, 1978.

[16]

G. F. Webb, Population models structured by age, size and position, in "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics, Vol. 1936, Springer-Verlag, Berlin-New York, (2008), 1-49.

[1]

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

[2]

Gabriella Di Blasio. Ultraparabolic equations with nonlocal delayed boundary conditions. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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]

Nguyen Thi Van Anh, Bui Thi Hai Yen. Source identification problems for abstract semilinear nonlocal differential equations. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022030

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Martin Burger, Jan-Frederik Pietschmann, Marie-Therese Wolfram. Identification of nonlinearities in transport-diffusion models of crowded motion. Inverse Problems and 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 and Continuous Dynamical Systems - S, 2016, 9 (3) : 777-790. doi: 10.3934/dcdss.2016028

[16]

Xiaohua Jing, Masahiro Yamamoto. Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022019

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]