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November  2013, 33(11&12): 4945-4965. doi: 10.3934/dcds.2013.33.4945

Ultraparabolic equations with nonlocal delayed boundary conditions

Gabriella Di Blasio 1,

1. 

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

Received  September 2011 Revised  February 2012 Published  May 2013

A class of ultraparabolic equations with delay, arising from age--structured population diffusion, is analyzed. For such equations well--posedness as well as regularity results with respect to the space variables are proved.
Keywords: population dynamics., Ultraparabolic equations, delay equations.
Mathematics Subject Classification: Primary: 35K70, 34K30; Secondary: 92D2.
Citation: 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
References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., 171 (1996), 41-62. doi: 10.1007/BF01759381.

[3]

B. E. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474. doi: 10.1006/jmaa.2000.6921.

[4]

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

[5]

L. I. Anita and S. Anita, Asymptotic behaviour of the solutions to semilinear age dependent population dynamics with diffusion and periodic vital rates, Math. Popul. Stud., 15 (2008), 114-122. doi: 10.1080/08898480802010175.

[6]

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

[7]

C. Cusulin, M. Iannelli and G. Marinoschi, Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real Word Appl., 6 (2005), 207-223. doi: 10.1016/j.nonrwa.2004.08.006.

[8]

G. Da Prato, "Applications Croissantes et Équations D' évolutions dans les Espaces de Banach," Institutiones Mathematicae II, Istituto Nazionale di Alta Matematica, Academic Press Inc., London, 1976.

[9]

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

[10]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7.

[11]

G. Di Blasio, Nonlinear age-dependent population diffusion, J. Math. Biol., 8 (1979), 265-284. doi: 10.1007/BF00276312.

[12]

G. Di Blasio, Mathematical analysis for an epidemic model with spatial and age structure, J. Evol. Equ., 10 (2010), 929-953. doi: 10.1007/s00028-010-0077-8.

[13]

A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251.

[14]

A. Ducrot and P. Magal, Travelling wave solutions for an infection age-structured model with diffusion, Proc. Roy. Soc. Edinburgh -A, 139 (2009), 2307-2325. doi: 10.1017/S0308210507000455.

[15]

J. Dyson, E. Sanchez, R. Villella-Bressan and G. F. Webb, An age and spatially structured model of tumor invasion with haptotaxis, II, Math. Popul. Stud., 15 (2008), 73-95. doi: 10.1080/08898480802010159.

[16]

G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behaviour of a population equation with diffusion and delayed birth process, Discrete Contin. Dynam. Systems - B, 7 (2007), 735-754. doi: 10.3934/dcdsb.2007.7.735.

[17]

M. E. Gurtin and R. C. MacCamy, Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.

[18]

K. Kunisch, W. Schappacher and G. F. Webb, Nonlinear age-dependent population dynamics with random diffusion, Comp. Math. Appl., 11 (1985), 155-173. doi: 10.1016/0898-1221(85)90144-0.

[19]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uralċeva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, AMS, Providence, 1968.

[20]

M. Langlais, A nonlinear problem in age-dependent population diffusion, SIAM J. Math. Anal., 16 (1985), 510-529. doi: 10.1137/0516037.

[21]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[22]

S. Piazzera, An age-dependent population equation with delayed birth process, Math. Methods Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.

[23]

S. Piazzera and L. Tonetto, Asynchronous exponential growth for an age-dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6.

[24]

X. Yu, Differentiability of an age-dependent population system with time delay in the birth process, J. Math. Anal. Appl., 303 (2005), 576-584. doi: 10.1016/j.jmaa.2004.08.061.

[25]

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

[26]

C. Walker, Global well-posedness of a haptotaxis model with spatial and age structure, Diff. Int. Eqs., 20 (2007), 1053-1074.

[27]

C. Walker, Age-dependent equations with nonlinear diffusion, Discrete Contin. Dynam. Systems - A, 26 (2010), 691-712. doi: 10.3934/dcds.2010.26.691.

[28]

G. F. Webb, Population models structured by age, size and position, in "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics 1936, Springer-Verlag, Berlin-New York, (2008), 1-49. doi: 10.1007/978-3-540-78273-5_1.

show all references

References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., 171 (1996), 41-62. doi: 10.1007/BF01759381.

[3]

B. E. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474. doi: 10.1006/jmaa.2000.6921.

[4]

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

[5]

L. I. Anita and S. Anita, Asymptotic behaviour of the solutions to semilinear age dependent population dynamics with diffusion and periodic vital rates, Math. Popul. Stud., 15 (2008), 114-122. doi: 10.1080/08898480802010175.

[6]

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

[7]

C. Cusulin, M. Iannelli and G. Marinoschi, Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real Word Appl., 6 (2005), 207-223. doi: 10.1016/j.nonrwa.2004.08.006.

[8]

G. Da Prato, "Applications Croissantes et Équations D' évolutions dans les Espaces de Banach," Institutiones Mathematicae II, Istituto Nazionale di Alta Matematica, Academic Press Inc., London, 1976.

[9]

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

[10]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7.

[11]

G. Di Blasio, Nonlinear age-dependent population diffusion, J. Math. Biol., 8 (1979), 265-284. doi: 10.1007/BF00276312.

[12]

G. Di Blasio, Mathematical analysis for an epidemic model with spatial and age structure, J. Evol. Equ., 10 (2010), 929-953. doi: 10.1007/s00028-010-0077-8.

[13]

A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Continuous Dynam. Systems - B, 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251.

[14]

A. Ducrot and P. Magal, Travelling wave solutions for an infection age-structured model with diffusion, Proc. Roy. Soc. Edinburgh -A, 139 (2009), 2307-2325. doi: 10.1017/S0308210507000455.

[15]

J. Dyson, E. Sanchez, R. Villella-Bressan and G. F. Webb, An age and spatially structured model of tumor invasion with haptotaxis, II, Math. Popul. Stud., 15 (2008), 73-95. doi: 10.1080/08898480802010159.

[16]

G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behaviour of a population equation with diffusion and delayed birth process, Discrete Contin. Dynam. Systems - B, 7 (2007), 735-754. doi: 10.3934/dcdsb.2007.7.735.

[17]

M. E. Gurtin and R. C. MacCamy, Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.

[18]

K. Kunisch, W. Schappacher and G. F. Webb, Nonlinear age-dependent population dynamics with random diffusion, Comp. Math. Appl., 11 (1985), 155-173. doi: 10.1016/0898-1221(85)90144-0.

[19]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uralċeva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, AMS, Providence, 1968.

[20]

M. Langlais, A nonlinear problem in age-dependent population diffusion, SIAM J. Math. Anal., 16 (1985), 510-529. doi: 10.1137/0516037.

[21]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[22]

S. Piazzera, An age-dependent population equation with delayed birth process, Math. Methods Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.

[23]

S. Piazzera and L. Tonetto, Asynchronous exponential growth for an age-dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6.

[24]

X. Yu, Differentiability of an age-dependent population system with time delay in the birth process, J. Math. Anal. Appl., 303 (2005), 576-584. doi: 10.1016/j.jmaa.2004.08.061.

[25]

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

[26]

C. Walker, Global well-posedness of a haptotaxis model with spatial and age structure, Diff. Int. Eqs., 20 (2007), 1053-1074.

[27]

C. Walker, Age-dependent equations with nonlinear diffusion, Discrete Contin. Dynam. Systems - A, 26 (2010), 691-712. doi: 10.3934/dcds.2010.26.691.

[28]

G. F. Webb, Population models structured by age, size and position, in "Structured Population Models in Biology and Epidemiology," Lecture Notes in Mathematics 1936, Springer-Verlag, Berlin-New York, (2008), 1-49. doi: 10.1007/978-3-540-78273-5_1.

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