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October  2019, 24(10): 5695-5707. doi: 10.3934/dcdsb.2019102

Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions

Anne Mund 1, , Christina Kuttler 1, and  Judith Pérez-Velázquez 1,2,,

1. 

Zentrum Mathematik, Technische Universität Müenchen, Boltzmannstr. 3, 85748 Garching, Germany
2. 

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Cd. de México

* Corresponding author: perez-velazquez@mym.iimas.unam.mx

Received  April 2018 Revised  November 2018 Published  October 2019 Early access  June 2019

We prove existence and uniqueness of weak and classical solutions to certain semi-linear parabolic systems with Robin boundary conditions using the coupled upper-lower solution approach. Our interest lies in cross-dependencies on the gradient parts of the reaction term, which prevents the straight-forward application of standard theorems. Such cross-dependencies emerge e.g. in a model describing evolution of bacterial quorum sensing, but are interesting also in a more general context. We show the existence and uniqueness of solutions for this example.

Keywords: Parabolic system, coupled upper-lower solutions, existence, uniqueness, semi-linear equation.
Mathematics Subject Classification: Primary: 35K51, 35B51; Secondary: 35K58.
Citation: Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
References:
[1]

M. Abudiab, I. Ahn and L. Li, Upper–lower solutions for nonlinear parabolic systems and their applications, Journal of Mathematical Analysis and Applications, 378 (2011), 620-633.  doi: 10.1016/j.jmaa.2011.01.003.  Google Scholar

[2]

D. Bothe, A. Fischer, M. Pierre and G. Rolland, Global well-posedness for a class of reaction–advection–anisotropic-diffusion systems, Journal of Evolution Equations, 17 (2017), 101-130.  doi: 10.1007/s00028-016-0348-0.  Google Scholar

[3]

K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, Journal of Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6.  Google Scholar

[4]

A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.  Google Scholar

[5]

N. Kronik and Y. Cohen, Evolutionary games in space, Mathematical Modelling of Natural Phenomena, 4 (2009), 54-90.  doi: 10.1051/mmnp/20094602.  Google Scholar

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O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968  Google Scholar

[7]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[8]

I. G. Petrovskii, On the Cauchy problem for systems of linear partial differential equations in the domain of non-analytic functions, Bull. MGU, Sect. A, 1938. Google Scholar

[9]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258 of Comprehensive Studies in Mathematics, Springer Verlag, 1983.  Google Scholar

[10]

D. Werner, Funktionalanalysis, Third, revised and extended edition. Springer-Verlag, Berlin, 2000.  Google Scholar

show all references

References:
[1]

M. Abudiab, I. Ahn and L. Li, Upper–lower solutions for nonlinear parabolic systems and their applications, Journal of Mathematical Analysis and Applications, 378 (2011), 620-633.  doi: 10.1016/j.jmaa.2011.01.003.  Google Scholar

[2]

D. Bothe, A. Fischer, M. Pierre and G. Rolland, Global well-posedness for a class of reaction–advection–anisotropic-diffusion systems, Journal of Evolution Equations, 17 (2017), 101-130.  doi: 10.1007/s00028-016-0348-0.  Google Scholar

[3]

K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, Journal of Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6.  Google Scholar

[4]

A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.  Google Scholar

[5]

N. Kronik and Y. Cohen, Evolutionary games in space, Mathematical Modelling of Natural Phenomena, 4 (2009), 54-90.  doi: 10.1051/mmnp/20094602.  Google Scholar

[6]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968  Google Scholar

[7]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[8]

I. G. Petrovskii, On the Cauchy problem for systems of linear partial differential equations in the domain of non-analytic functions, Bull. MGU, Sect. A, 1938. Google Scholar

[9]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258 of Comprehensive Studies in Mathematics, Springer Verlag, 1983.  Google Scholar

[10]

D. Werner, Funktionalanalysis, Third, revised and extended edition. Springer-Verlag, Berlin, 2000.  Google Scholar

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