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

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  • 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.

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

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