Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. (English) Zbl 1420.35119
Summary: 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.
MSC:
35K51 | Initial-boundary value problems for second-order parabolic systems |
35B51 | Comparison principles in context of PDEs |
35K58 | Semilinear parabolic equations |