  In this paper we provide an example of a class of two reaction-diffusion-ODE
equations with homogeneous Neumann boundary conditions, in which Turing-type
instability not only destabilizes constant steady states but also induces
blow-up of nonnegative spatially heterogeneous solutions. Solutions of this
problem preserve nonnegativity and uniform boundedness of the total mass.
Moreover, for the corresponding system with two non-zero diffusion
coefficients, all nonnegative solutions are global in time. We prove that a
removal of diffusion in one of the equations leads to a finite-time blow-up of
some nonnegative spatially heterogeneous solutions.
