American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 733-744. doi: 10.3934/proc.2015.0733

Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model

Akisato Kubo 1, , Hiroki Hoshino 2, and  Katsutaka Kimura 2,

1. 

Department of Mathematics, School of Health Sciences, Fujita Health University, Toyoake, Aichi 470-1192
2. 

School of Health Sciences, Fujita Health University, Toyoake, Aichi 470-1192, Japan, Japan

Received  September 2014 Revised  January 2015 Published  November 2015

We study the global existence in time and asymptotic behaviour of solutions of nonlinear evolution equations with strong dissipation and proliferation terms arising in mathematical models of biology and medicine including tumour invasion models.
Keywords: energy estimate, mathematical model., asymptotic behaviour, time global solution, tumour invasion, Nonlinear evolution equation.
Mathematics Subject Classification: Primary: 35L53; Secondary: 35K1.
Citation: Akisato Kubo, Hiroki Hoshino, Katsutaka Kimura. Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model. Conference Publications, 2015, 2015 (special) : 733-744. doi: 10.3934/proc.2015.0733
References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett., 11 (1998), 109-114.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.

[3]

M. A. J. Chaplain and G. Lolas, Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.

[4]

B. Davis, Reinforced random walks, Probability Theory and Related Fields, 84 (1990), 203-229.

[5]

P. Dionne, Sur les problemes de Cauchy hyperboliques bien poses, J. Anal. Math., 10 (1962), 1-90.

[6]

Y. Ebihara, On some nonlinear evolution equations with the strong dissipation, J. Differential Equations, 30 (1978), 149-164.

[7]

Y. Ebihara, On some nonlinear evolution equations with the strong dissipation, II, J. Differential Equations, 34 (1979), 339-352.

[8]

Y. Ebihara, On some nonlinear evolution equations with strong dissipation, III, J. Differential Equations, 45 (1982), 332-355.

[9]

A. Kubo, Nonlinear evolution equations associated with mathematical models, Discrete and Continuous Dynamical Systems supplement 2011, (2011), 881-890.

[10]

A. Kubo and T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential and Integral Equations, 17 (2004), 721-736.

[11]

A. Kubo, T. Suzuki and H. Hoshino, Asymptotic behavior of the solution to a parabolic ODE system, Mathematical Sciences and Applications, 22 (2005), 121-135.

[12]

A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55.

[13]

A. Kubo, N. Saito, T. Suzuki and H. Hoshino, Mathematical models of tumour angiogenesis and simulations, Theory of Bio-Mathematics and Its Applications, in RIMS Kokyuroku, 1499 (2006), 135-146.

[14]

H. A. Levine and B. D. Sleeman, A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.

[15]

S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press. London, (1973).

[16]

B. D. Sleeman and H.A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Mech. Appl. Sci., 24 (2001), 405-426.

[17]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.

[18]

A. Kubo and H. Hoshino, Nonlinear evolution equations with strong dissipation and proliferation, Current Trends in Analysis and Applications, 233-241, Birkhauser, Springer, 2015.

show all references

References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett., 11 (1998), 109-114.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.

[3]

M. A. J. Chaplain and G. Lolas, Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.

[4]

B. Davis, Reinforced random walks, Probability Theory and Related Fields, 84 (1990), 203-229.

[5]

P. Dionne, Sur les problemes de Cauchy hyperboliques bien poses, J. Anal. Math., 10 (1962), 1-90.

[6]

Y. Ebihara, On some nonlinear evolution equations with the strong dissipation, J. Differential Equations, 30 (1978), 149-164.

[7]

Y. Ebihara, On some nonlinear evolution equations with the strong dissipation, II, J. Differential Equations, 34 (1979), 339-352.

[8]

Y. Ebihara, On some nonlinear evolution equations with strong dissipation, III, J. Differential Equations, 45 (1982), 332-355.

[9]

A. Kubo, Nonlinear evolution equations associated with mathematical models, Discrete and Continuous Dynamical Systems supplement 2011, (2011), 881-890.

[10]

A. Kubo and T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential and Integral Equations, 17 (2004), 721-736.

[11]

A. Kubo, T. Suzuki and H. Hoshino, Asymptotic behavior of the solution to a parabolic ODE system, Mathematical Sciences and Applications, 22 (2005), 121-135.

[12]

A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55.

[13]

A. Kubo, N. Saito, T. Suzuki and H. Hoshino, Mathematical models of tumour angiogenesis and simulations, Theory of Bio-Mathematics and Its Applications, in RIMS Kokyuroku, 1499 (2006), 135-146.

[14]

H. A. Levine and B. D. Sleeman, A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.

[15]

S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press. London, (1973).

[16]

B. D. Sleeman and H.A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Mech. Appl. Sci., 24 (2001), 405-426.

[17]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.

[18]

A. Kubo and H. Hoshino, Nonlinear evolution equations with strong dissipation and proliferation, Current Trends in Analysis and Applications, 233-241, Birkhauser, Springer, 2015.

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