A note on the boundedness of solutions of linear parabolic equations
Alan McNabb · Proceedings of the American Mathematical Society · 1962
in a bounded domain B with boundary 9B. In this note, their method is slightly modified to prove the following theorem. Denote by D the semi-infinite cylinder I (x, t): xEB, t > } }, by D its closure and by DT the intersection of D with the half-space t ?T. Suppose u = u(x) is a solution of equation (1) which is continuous in B the closure of B, vanishes on 9B and has continuous second derivatives in B. Again, suppose w= w(x, t) is defined and continuous in the closed region DT, is positive on B at t = 0 and on 9B for all t 0 O and has continuous derivatives Ow/Ot, O2W/OxiOxX which satisfy the parabolic equation