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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

 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  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.

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
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