In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results.

We consider the problem of computing numerically the boundary control for the wave equation. It is by now well known that, due to high frequency spurious oscillations, numerical instabilities occur and may led to the failure of convergence of some apparently natural numerical algorithms. Several remedies have been proposed in the literature to compensate this fact: Tychonoff regularization, Fourier filtering, mixed finite elements,… In this Note we prove that the two-grid method proposed by Glowinski (J. Comput. Phys. 103 (2) (1992) 189–221) does indeed provide a convergent algorithm. This is done in the context of the finite-difference semi-discrete approximation of the 1-d wave equation.