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Ultraparabolic equations with nonlocal delayed boundary conditions
1. | Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 5, Roma, 00185, Italy |
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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