Heat-type equations on manifolds with fibered boundaries II: Parametrix construction

This is the second part of a two parts work on the analysis of heat-type equations on manifolds with fibered boundary equipped with a $\mathrm{\Phi }$-metric. This setting generalizes the asymptotically conical (scattering) spaces and includes special cases of magnetic and gravitational monopoles. The core of this second part consists on the construction of parametrix for heat-type equations. Consequently we use the constructed parametrix to infer results regarding existence and regularity of certain homogeneous and non homogeneous second order linear parabolic equations with non constant coefficients. This work represents the first step towards the analysis of geometric flows such as Ricci-, Yamabe and Mean Curvature flow on some families of non compact manifolds.