Two-dimensional transonic flows which are bounded by a free surface and a wall and which have interior heat release are investigated theoretically. It is shown that under certain conditions a supersonic-subsonic transition is possible, in contrast to flows bounded by two walls, while maintaining a positive pressure coefficient on the free surface when the free surface is formed by an adjacent supersonic or transonic flow. The problems investigated correspond to outer limits in the sense of matched asymptotic expansions and arbitrary initial conditions cannot be satisfied; however, several practical problems yield initial conditions which fortuitously match the outer limit conditions. Application to problems of external burning for control or propulsion is indicated.