On a nonlinear system of PDE’s arising in free convection

In this paper, we consider a model problem introduced in [2], and derived from a coupled system of partial differential equation arising in the study of free convection about a vertical flat plate embedded in a porous medium. In [2], some existence result has been obtained under very constraining hypothesis. Here, we come back to the weak formulation of this problem, and prove that there is one and only one weak solution, under reasonable hypotheses on the data. Details about the physical background can be found, for example, in [13], [14], [15], [16], [19] and [22]. In these papers, the authors assume that convection takes place in a thin layer around the plate. This allows to make boundary-layer approximations, and to get similarity solutions by solving the ordinary differential equation