In this contribution, we propose to generate adaptive space-time meshes to capture fronts. As a toy problem, we use the Porous Medium Equation (PME) that is a non-linear partial differential equation of significant interest due to its widespread applicability in modelling various physical phenomena such as gas flow in porous medium, incompressible fluid dynamics and non-linear heat transfer. The nature of the solutions to the PME differs fundamentally depending on whether we consider the linear case or the non-linear case. In the non-linear scenario, if the initial distribution u0(x) of the quantity of interest u has a compact support, then the solution u(x, t) will have a compact support for t > 0. There thus exist a front that separates the part of the domain where u>0 and the other part where u=0. In this paper, we compute The PME using a space-time formulation. We focus on the 2D computation in space plus time. In each time step, we use the relaying technique proposed recently to capture the mesh in space. Tetrahedra connect the different spatial planes and the tetrahedral mesh is adapted to capture the exact front in space and time, leading to a mimetic representation of the front.