A chemotaxis model motivated by angiogenesis

We consider a simple model arising in modeling angiogenesis and more specifically the development of capillary blood vessels due to an exogenous chemo-attractive signal (solid tumors for instance). It is given as coupled system of parabolic equations through a nonlinear transport term. We show that, by opposition to some classical chemotaxis model, this system admits a positive energy. This allows us to develop an existence theory for weak solutions. We also show that, in two dimensions, this system admits a family of self-similar waves.