Convection under certain continents should be strongly affected by the presence of a neighboring subduction zone. The cold subducting oceanic lithosphere is removing heat from the continental mantle. Model cavities with lateral cooling are investigated to quantify the physics of such situations. Convection cells with a large aspect ratio are generated, and a sizable portion, typically 50% or more, of the heat produced under continents is transferred to the oceanic lithosphere in the models. This leads to geophysical speculations concerning the global heat balance in the earth and the capability of large subcontinental cells to drive the continental plates towards the trenches. Other points are discussed, particularly the influence of the chosen rheology (constant viscosity, Herring-Nabarro or Weertman creep) and the reduced heat flow profiles.