May  2020, 13(5): 1473-1493. doi: 10.3934/dcdss.2020083

Global existence for Laplace reaction-diffusion equations

1. 

Department of Mathematics, Università degli Studi di Bologna, piazza di Porta S. Donato 5, 40126 Bologna, Italy

2. 

Professor Emeritus, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan

* Corresponding author: Atsushi Yagi

Received  June 2018 Revised  July 2018 Published  June 2019

We study the initial-boundary value problem for a Laplace reaction-diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show global existence under suitable assumptions on the reaction function. We also show that the problem generates a dynamical system in a suitably set universal space and that this dynamical system possesses a Lyapunov function.

Citation: Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
References:
[1]

A. A. Amosov and O. A. Amosova, Error estimates for FEM schemes constructed for the degenerate diffusion equations with discontinuous coefficients, Soviet J. Mumer. Anal. Math. Modelling, 1 (1986), 163-187.   Google Scholar

[2]

H. Carslaw and J. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988.  Google Scholar

[3]

F. de Monte, Transient heat conduction in one-dimensional composite slab. A 'natural' analytic approach, Inter. J. Heat Mass Trans., 43 (2000), 3607-3619.  doi: 10.1016/S0017-9310(00)00008-9.  Google Scholar

[4]

B. Deconinck, B. Pelloni and N. E. Sheils, Non-steady state heat conduction in composite walls, Proc. Royal Soc. A, 470 (2014), 20130605. doi: 10.1098/rspa.2013.0605.  Google Scholar

[5]

A. Favini and A. Yagi, Space and time regularity for degenerate evolution equations, J. Math. Soc. Japan, 44 (1992), 331-350.  doi: 10.2969/jmsj/04420331.  Google Scholar

[6]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., (Ⅳ) 163 (1993), 353–384. doi: 10.1007/BF01759029.  Google Scholar

[7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, CRC Press, 1999.   Google Scholar
[8]

D. W. Hahn and M. N. Özişik, Heart Conduction, John Wiley & Sons Inc., 2012. Google Scholar

[9]

X -M. HeT. Lin and Y. Lin, A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficients, Commun. Comput. Phys., 6 (2009), 185-202.  doi: 10.4208/cicp.2009.v6.p185.  Google Scholar

[10]

Gh. Juncu and C. Popa, Preconditioning by Gram matrix approximation for diffusion-convection-reaction equations with discontinuous coefficients, Math. Comput. Simulation, 60 (2002), 487-506.  doi: 10.1016/S0378-4754(02)00063-0.  Google Scholar

[11]

D. Kim, Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 55–76.  Google Scholar

[12]

L. V. Korobenko and V. Zh. Sakbaev, On the formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerate coefficients, Comput. Math. Math. Phys., 49 (2009), 1037-1053.  doi: 10.1134/S0965542509060128.  Google Scholar

[13]

F. LemariéL. Debreu and E. Blayo, Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients, Electron. Trans. Numer. Anal., 40 (2013), 148-186.   Google Scholar

[14]

M. D. Mikhailov and M. N. Özişik, Transient conduction in a three-dimensional composite slab, Inter. J. Heat and Mass Transfer, 29 (1986), 340-342.  doi: 10.1016/0017-9310(86)90242-5.  Google Scholar

[15]

H. Salt, Transient conduction in a two-dimensional composite slab, Inter. J. Heat and Mass Transfer, 26 (1983), 1611-1623.   Google Scholar

[16]

N. E. Sheils and B. Deconinck, Initial-to-interface maps for the heat equation on composite domains, Stud. Appl. Math., 137 (2016), 140-154.  doi: 10.1111/sapm.12138.  Google Scholar

[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

show all references

References:
[1]

A. A. Amosov and O. A. Amosova, Error estimates for FEM schemes constructed for the degenerate diffusion equations with discontinuous coefficients, Soviet J. Mumer. Anal. Math. Modelling, 1 (1986), 163-187.   Google Scholar

[2]

H. Carslaw and J. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988.  Google Scholar

[3]

F. de Monte, Transient heat conduction in one-dimensional composite slab. A 'natural' analytic approach, Inter. J. Heat Mass Trans., 43 (2000), 3607-3619.  doi: 10.1016/S0017-9310(00)00008-9.  Google Scholar

[4]

B. Deconinck, B. Pelloni and N. E. Sheils, Non-steady state heat conduction in composite walls, Proc. Royal Soc. A, 470 (2014), 20130605. doi: 10.1098/rspa.2013.0605.  Google Scholar

[5]

A. Favini and A. Yagi, Space and time regularity for degenerate evolution equations, J. Math. Soc. Japan, 44 (1992), 331-350.  doi: 10.2969/jmsj/04420331.  Google Scholar

[6]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., (Ⅳ) 163 (1993), 353–384. doi: 10.1007/BF01759029.  Google Scholar

[7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, CRC Press, 1999.   Google Scholar
[8]

D. W. Hahn and M. N. Özişik, Heart Conduction, John Wiley & Sons Inc., 2012. Google Scholar

[9]

X -M. HeT. Lin and Y. Lin, A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficients, Commun. Comput. Phys., 6 (2009), 185-202.  doi: 10.4208/cicp.2009.v6.p185.  Google Scholar

[10]

Gh. Juncu and C. Popa, Preconditioning by Gram matrix approximation for diffusion-convection-reaction equations with discontinuous coefficients, Math. Comput. Simulation, 60 (2002), 487-506.  doi: 10.1016/S0378-4754(02)00063-0.  Google Scholar

[11]

D. Kim, Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 55–76.  Google Scholar

[12]

L. V. Korobenko and V. Zh. Sakbaev, On the formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerate coefficients, Comput. Math. Math. Phys., 49 (2009), 1037-1053.  doi: 10.1134/S0965542509060128.  Google Scholar

[13]

F. LemariéL. Debreu and E. Blayo, Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients, Electron. Trans. Numer. Anal., 40 (2013), 148-186.   Google Scholar

[14]

M. D. Mikhailov and M. N. Özişik, Transient conduction in a three-dimensional composite slab, Inter. J. Heat and Mass Transfer, 29 (1986), 340-342.  doi: 10.1016/0017-9310(86)90242-5.  Google Scholar

[15]

H. Salt, Transient conduction in a two-dimensional composite slab, Inter. J. Heat and Mass Transfer, 26 (1983), 1611-1623.   Google Scholar

[16]

N. E. Sheils and B. Deconinck, Initial-to-interface maps for the heat equation on composite domains, Stud. Appl. Math., 137 (2016), 140-154.  doi: 10.1111/sapm.12138.  Google Scholar

[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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