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 & Continuous Dynamical Systems - A, 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. doi: 10.3934/cpaa.2013.12.2069. Google Scholar

[2]

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

[3]

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

[4]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, Abstr. Appl. Anal., (4570). doi: 10.1155/2012/457019. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[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. doi: 10.1142/S0218127410027489. Google Scholar

[10]

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

[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. doi: 10.1016/j.jfa.2006.12.008. Google Scholar

[12]

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

[13]

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

[14]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, With contributions by S. Brendle, (2000). Google Scholar

[15]

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

[16]

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

[17]

K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011). doi: 10.1007/978-1-4471-2170-1. Google Scholar

[18]

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

[19]

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

[20]

R. W. King and G. S. Smith, Antennas in Matter: Fundamental, Theory and Application,, MIT Press, (1981). Google Scholar

[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. doi: 10.1088/0031-9155/53/5/018. Google Scholar

[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). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[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. doi: 10.1093/imammb/dqp002. Google Scholar

[28]

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

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. doi: 10.3934/cpaa.2013.12.2069. Google Scholar

[2]

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

[3]

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

[4]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, Abstr. Appl. Anal., (4570). doi: 10.1155/2012/457019. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[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. doi: 10.1142/S0218127410027489. Google Scholar

[10]

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

[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. doi: 10.1016/j.jfa.2006.12.008. Google Scholar

[12]

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

[13]

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

[14]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, With contributions by S. Brendle, (2000). Google Scholar

[15]

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

[16]

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

[17]

K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011). doi: 10.1007/978-1-4471-2170-1. Google Scholar

[18]

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

[19]

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

[20]

R. W. King and G. S. Smith, Antennas in Matter: Fundamental, Theory and Application,, MIT Press, (1981). Google Scholar

[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. doi: 10.1088/0031-9155/53/5/018. Google Scholar

[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). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[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. doi: 10.1093/imammb/dqp002. Google Scholar

[28]

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

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