Behavior of the solution of a parabolic equationon a characteristic

We consider the solution of a linear second-order parabolic equation with one spatial variable and a zero right side. We prove that since the solution decreases quite rapidly in the spatial variable as it approaches a particular point, it vanishes on the part of the characteristic joining the point to the boundary of the region in which the solution is defined.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 257–262, September, 1972.