American Institute of Mathematical Sciences

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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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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. Google Scholar

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