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

Ultraparabolic equations with nonlocal delayed boundary conditions

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.
Citation: 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
References:
[1]

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

[2]

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

[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. doi: 10.1006/jmaa.2000.6921. Google Scholar

[4]

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

[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. doi: 10.1080/08898480802010175. Google Scholar

[6]

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

[7]

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

[8]

G. Da Prato, "Applications Croissantes et Équations D' évolutions dans les Espaces de Banach,", Institutiones Mathematicae II, (1976). Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1017/S0308210507000455. Google Scholar

[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. doi: 10.1080/08898480802010159. Google Scholar

[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. doi: 10.3934/dcdsb.2007.7.735. Google Scholar

[17]

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

[18]

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

[19]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uralċeva, "Linear and Quasilinear Equations of Parabolic Type,", Transl. Math. Monographs, (1968). Google Scholar

[20]

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

[21]

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

[22]

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

[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. doi: 10.1007/s00028-004-0159-6. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

G. F. Webb, Population models structured by age, size and position,, in, 1936 (2008), 1. doi: 10.1007/978-3-540-78273-5_1. 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. doi: 10.1007/BF02774019. Google Scholar

[2]

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

[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. doi: 10.1006/jmaa.2000.6921. Google Scholar

[4]

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

[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. doi: 10.1080/08898480802010175. Google Scholar

[6]

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

[7]

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

[8]

G. Da Prato, "Applications Croissantes et Équations D' évolutions dans les Espaces de Banach,", Institutiones Mathematicae II, (1976). Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1017/S0308210507000455. Google Scholar

[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. doi: 10.1080/08898480802010159. Google Scholar

[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. doi: 10.3934/dcdsb.2007.7.735. Google Scholar

[17]

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

[18]

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

[19]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uralċeva, "Linear and Quasilinear Equations of Parabolic Type,", Transl. Math. Monographs, (1968). Google Scholar

[20]

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

[21]

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

[22]

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

[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. doi: 10.1007/s00028-004-0159-6. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

G. F. Webb, Population models structured by age, size and position,, in, 1936 (2008), 1. doi: 10.1007/978-3-540-78273-5_1. Google Scholar

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