Summary: We consider non-negative solution couples \((u,v)\) of \[ \begin{cases} u_t=u_{xx}-\chi(\frac uvv_x)_x-\lambda u,\\ v_t=1-v+u,\end{cases} \] with positive parameters \(\chi\) and \(\lambda\), where the spatial domain is the interval \((0,1)\). This system appears as a limit case of a model for morphogenesis proposed by T. Bollenbach et al. [“Morphogen transport in epithelia”, Phys. Rev. E 75, No. 1, Article ID 011901, 16 p. (2007; doi:10.1103/PhysRevE.75.011901)]. Under suitable boundary conditions, modeling the presence of a morphogen source at \(x=0\), we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover, we prove the convergence of the solution to the unique steady state provided that \(\chi\) is small and \(\lambda\) is large enough.
Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements.