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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 & 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.  Google Scholar

[2]

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

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics," Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publisher, Dordrecht, 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., 37 (2000), 1571-1589.  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), 305-387.  Google Scholar

[8]

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

[9]

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

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273.  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), 45-60.  Google Scholar

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics, J. Evol. Equ., 2 (2002), 223-239.  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), 663-683.  Google Scholar

[14]

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

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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 "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics, Vol. 1936, Springer-Verlag, Berlin-New York, (2008), 1-49.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

S. Anita, "Analysis and Control of Age-Dependent Population Dynamics," Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publisher, Dordrecht, 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., 37 (2000), 1571-1589.  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), 305-387.  Google Scholar

[8]

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

[9]

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

[10]

A. Ducrot, Travelling wave solutions fo a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273.  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), 45-60.  Google Scholar

[12]

M. Gyllenberg, A. Osipov and L. Päivärinta, The inverse problem for linear age-structured population dynamics, J. Evol. Equ., 2 (2002), 223-239.  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), 663-683.  Google Scholar

[14]

W. Rundell, Determining the death rate for an age-structured population from census data, SIAM J. Appl. Math., 53 (1993), 1731-1746.  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 "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics, Vol. 1936, Springer-Verlag, Berlin-New York, (2008), 1-49.  Google Scholar

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