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A note on the boundedness of solutions of linear parabolic equations

A. McNabb
1962 Proceedings of the American Mathematical Society
Hartman and Wintner [l] obtained a Sturmian comparison theorem for self-adjoint second order elliptic equations of the form (1) Z -( aa-) in a bounded domain B with boundary dB. In this note, their method is slightly modified to prove the following theorem. Denote by D the semi-infinite cylinder {(x, /):xG-B,/>0},by 7) its closure and by Dt the intersection of D with the half-space / ^ T. Suppose u = u(x) is a solution of equation (1) which is continuous in B the closure of B, vanishes on dB
more » ... , vanishes on dB 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 / = 0 and on dB for all / ^ 0 and has continuous derivatives dw/dt, d2w/dxidx¡ which satisfy the parabolic equation in Dt for all 7>0. The functions An, dAn/dxi, Fand Care uniformly bounded continuous functions of Xt and / in DT for any given 7">0, while C is bounded in D by two positive constants G and G (00 and fSe(t)dt tends to infinity with T, then w is unbounded in D. Proof. Since w>0 on .B at / = 0 and on dB for all />0, the maximum principle for parabolic equations (see [2]) implies w>0 in Dt for all 7>0. The Green identity leads from the boundary condition t< = 0 on dB, to the divergence relation (3)