Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis

Angiogenesis is a key process in the tumoral growth which allows the cancerous tissue to impact on its vasculature in order to improve the nutrient’s supply and the metastatic process. In this paper, we introduce a model for the density of metastasis which takes into account for this feature. It is a two-dimensional structured population equation with a vanishing velocity field and a source term on the boundary. We present here the mathematical analysis of the model, namely the well-posedness of the equation and the asymptotic behavior of the solutions, whose natural regularity led us to investigate some basic properties of the space W_div(W)={V ? L¹;  div(GV) ? L¹}{W_{\rm div}(\Omega)=\left\{V\in L^1;{\rm div}(GV)\in L^1\right\}}, where G is the velocity field of the equation.