Article Contents

2019, Volume 24, Issue 10: 5695-5707. Doi: 10.3934/dcdsb.2019102

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

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

Received: April 2018

Revised: November 2018

Early access: June 2019

Published: August 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K51, 35B51; Secondary: 35K58.

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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.
[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.
[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.
[4]	A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.
[5]	N. Kronik and Y. Cohen, Evolutionary games in space, Mathematical Modelling of Natural Phenomena, 4 (2009), 54-90. doi: 10.1051/mmnp/20094602.
[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
[7]	J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144. doi: 10.1137/0520075.
[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.
[9]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258 of Comprehensive Studies in Mathematics, Springer Verlag, 1983.
[10]	D. Werner, Funktionalanalysis, Third, revised and extended edition. Springer-Verlag, Berlin, 2000.