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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 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.

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