American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., 171 (1996), 41-62. 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), 455-474. doi: 10.1006/jmaa.2000.6921.  Google Scholar

[4]

S. Anita, "Analysis and Control of Age-dependent Population Dynamics," Mathematical Modelling: Theory and Applications, Kluwer Academic Publisher, Dordrecht, 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), 114-122. doi: 10.1080/08898480802010175.  Google Scholar

[6]

P. L. Butzer and H. Berens, "Semi-Groups of Operators and Approximation," Springer-Verlag, Berlin, 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), 207-223. 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, Istituto Nazionale di Alta Matematica, Academic Press Inc., London, 1976.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

G. Di Blasio, Nonlinear age-dependent population diffusion, J. Math. Biol., 8 (1979), 265-284. 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), 929-953. 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), 251-273. 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), 2307-2325. 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), 73-95. 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), 735-754. 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), 281-300.  Google Scholar

[18]

K. Kunisch, W. Schappacher and G. F. Webb, Nonlinear age-dependent population dynamics with random diffusion, Comp. Math. Appl., 11 (1985), 155-173. 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, AMS, Providence, 1968.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  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), 61-77. 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), 576-584. 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), 1053-1074.  Google Scholar

[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.  Google Scholar

[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.  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. doi: 10.1007/BF02774019.  Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura e Appl., 171 (1996), 41-62. 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), 455-474. doi: 10.1006/jmaa.2000.6921.  Google Scholar

[4]

S. Anita, "Analysis and Control of Age-dependent Population Dynamics," Mathematical Modelling: Theory and Applications, Kluwer Academic Publisher, Dordrecht, 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), 114-122. doi: 10.1080/08898480802010175.  Google Scholar

[6]

P. L. Butzer and H. Berens, "Semi-Groups of Operators and Approximation," Springer-Verlag, Berlin, 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), 207-223. 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, Istituto Nazionale di Alta Matematica, Academic Press Inc., London, 1976.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

G. Di Blasio, Nonlinear age-dependent population diffusion, J. Math. Biol., 8 (1979), 265-284. 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), 929-953. 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), 251-273. 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), 2307-2325. 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), 73-95. 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), 735-754. 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), 281-300.  Google Scholar

[18]

K. Kunisch, W. Schappacher and G. F. Webb, Nonlinear age-dependent population dynamics with random diffusion, Comp. Math. Appl., 11 (1985), 155-173. 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, AMS, Providence, 1968.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  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), 61-77. 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), 576-584. 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), 1053-1074.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
[2]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[3]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[4]

Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
[5]

Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768
[6]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[7]

Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541
[8]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256
[9]

King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051
[10]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[11]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[12]

Emile Franc Doungmo Goufo. Bounded perturbation for evolution equations with a parameter & application to population dynamics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2137-2150. doi: 10.3934/dcdss.2020177
[13]

Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395
[14]

Gabriella Di Blasio. An ultraparabolic problem arising from age-dependent population diffusion. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 843-858. doi: 10.3934/dcds.2009.25.843
[15]

Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362
[16]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051
[17]

Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071
[18]

Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735
[19]

Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32
[20]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy