February  2015, 35(2): 653-668. doi: 10.3934/dcds.2015.35.653

Chaos for the Hyperbolic Bioheat Equation

1. 

Dept. Matemàtica Aplicada and IUMPA, Universitat Politècnica de València, València, 46022, Spain, Spain, Spain

Received  July 2013 Revised  January 2014 Published  September 2014

The Hyperbolic Heat Transfer Equation describes heat processes in which extremely short periods of time or extreme temperature gradients are involved. It is already known that there are solutions of this equation which exhibit a chaotic behaviour, in the sense of Devaney, on certain spaces of analytic functions with certain growth control. We show that this chaotic behaviour still appears when we add a source term to this equation, i.e. in the Hyperbolic Bioheat Equation. These results can also be applied for the Wave Equation and for a higher order version of the Hyperbolic Bioheat Equation.
Citation: J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
References:
[1]

A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal., 12 (2013), 2069-2082. doi: 10.3934/cpaa.2013.12.2069.

[2]

J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.

[3]

J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79. doi: 10.3934/dcds.2011.29.67.

[4]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations, Abstr. Appl. Anal., Art. ID 457019, 11 pp. doi: 10.1155/2012/457019.

[5]

T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. doi: 10.1016/j.jmaa.2010.06.011.

[6]

T. Bermúdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math., 170 (2005), 57-75.

[7]

N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal., 265 (2013), 2143-2163. doi: 10.1016/j.jfa.2013.06.019.

[8]

Z. Brzeźniak and A. L. Dawidowicz, On periodic solutions to the von Foerster-Lasota equation, Semigroup Forum, 78 (2009), 118-137. doi: 10.1007/s00233-008-9120-2.

[9]

J. A. Conejero, A. Peris and M. Trujillo, Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2943-2947. doi: 10.1142/S0218127410027489.

[10]

J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109. doi: 10.1007/s00009-010-0030-7.

[11]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.

[12]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.

[13]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.

[14]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[15]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.

[16]

J. Greer, A. Bertozzi and G. Shapiro, Fourth order partial differential equations on general geometries, Journal of Computational Physics, 216 (2006), 216-246. doi: 10.1016/j.jcp.2005.11.031.

[17]

K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.

[18]

G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171. doi: 10.1002/mana.19971880110.

[19]

T. Kalmes, On chaotic $C_0$-semigroups and infinitely regular hypercyclic vectors, Proc. Amer. Math. Soc., 134 (2006), 2997-3002 (electronic). doi: 10.1090/S0002-9939-06-08391-2.

[20]

R. W. King and G. S. Smith, Antennas in Matter: Fundamental, Theory and Application, MIT Press, Cambridge, MA, 1981.

[21]

J. A. López-Molina, M. J. Rivera, M. Trujillo and E. J. Berjano, Effect of the thermal wave in radiofrequency ablation modeling: An analytical study, Phys. Med. Biol., 53 (2008), 1447-1462. doi: 10.1088/0031-9155/53/5/018.

[22]

J. A. López-Molina, M. J. Rivera and E. J. Berjano, Fourier, hyperbolic and relativistic heat transfer equations for mathematical modeling of radiofrequency ablation of biological tissues: A comparative analytical study, preprint, (2013).

[23]

T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.

[24]

M. Özisik and D. Y. Tzou, On the wave theory on heat conduction, J. Heat Transfer, 116 (1994), 526-535. doi: 10.1115/1.2910903.

[25]

H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm, 1948, J. Appl. Physiol., 85 (1998), 5-34.

[26]

N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73.

[27]

M. Trujillo, M. J. Rivera, J. A. López-Molina and E. J. Berjano, Analytical thermal-optic model for laser heating of biological tissue using the hyperbolic heat transfer equation, Math. Med. Biol., 26 (2009), 187-200. doi: 10.1093/imammb/dqp002.

[28]

A. J. Welch and M. J. Van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, Plenum, New York, 1995. doi: 10.1007/978-1-4757-6092-7.

show all references

References:
[1]

A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal., 12 (2013), 2069-2082. doi: 10.3934/cpaa.2013.12.2069.

[2]

J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.

[3]

J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79. doi: 10.3934/dcds.2011.29.67.

[4]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations, Abstr. Appl. Anal., Art. ID 457019, 11 pp. doi: 10.1155/2012/457019.

[5]

T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. doi: 10.1016/j.jmaa.2010.06.011.

[6]

T. Bermúdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math., 170 (2005), 57-75.

[7]

N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal., 265 (2013), 2143-2163. doi: 10.1016/j.jfa.2013.06.019.

[8]

Z. Brzeźniak and A. L. Dawidowicz, On periodic solutions to the von Foerster-Lasota equation, Semigroup Forum, 78 (2009), 118-137. doi: 10.1007/s00233-008-9120-2.

[9]

J. A. Conejero, A. Peris and M. Trujillo, Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2943-2947. doi: 10.1142/S0218127410027489.

[10]

J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109. doi: 10.1007/s00009-010-0030-7.

[11]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.

[12]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.

[13]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.

[14]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[15]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.

[16]

J. Greer, A. Bertozzi and G. Shapiro, Fourth order partial differential equations on general geometries, Journal of Computational Physics, 216 (2006), 216-246. doi: 10.1016/j.jcp.2005.11.031.

[17]

K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.

[18]

G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171. doi: 10.1002/mana.19971880110.

[19]

T. Kalmes, On chaotic $C_0$-semigroups and infinitely regular hypercyclic vectors, Proc. Amer. Math. Soc., 134 (2006), 2997-3002 (electronic). doi: 10.1090/S0002-9939-06-08391-2.

[20]

R. W. King and G. S. Smith, Antennas in Matter: Fundamental, Theory and Application, MIT Press, Cambridge, MA, 1981.

[21]

J. A. López-Molina, M. J. Rivera, M. Trujillo and E. J. Berjano, Effect of the thermal wave in radiofrequency ablation modeling: An analytical study, Phys. Med. Biol., 53 (2008), 1447-1462. doi: 10.1088/0031-9155/53/5/018.

[22]

J. A. López-Molina, M. J. Rivera and E. J. Berjano, Fourier, hyperbolic and relativistic heat transfer equations for mathematical modeling of radiofrequency ablation of biological tissues: A comparative analytical study, preprint, (2013).

[23]

T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.

[24]

M. Özisik and D. Y. Tzou, On the wave theory on heat conduction, J. Heat Transfer, 116 (1994), 526-535. doi: 10.1115/1.2910903.

[25]

H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm, 1948, J. Appl. Physiol., 85 (1998), 5-34.

[26]

N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73.

[27]

M. Trujillo, M. J. Rivera, J. A. López-Molina and E. J. Berjano, Analytical thermal-optic model for laser heating of biological tissue using the hyperbolic heat transfer equation, Math. Med. Biol., 26 (2009), 187-200. doi: 10.1093/imammb/dqp002.

[28]

A. J. Welch and M. J. Van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, Plenum, New York, 1995. doi: 10.1007/978-1-4757-6092-7.

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