The uniqueness of solutions of the heat equation in an infinite strip
Victor L. Shapiro · Transactions of the American Mathematical Society · 1966
THEOREM 1. Let u(x, t) be a solution of the heat equation in the strip 0 0; (ii) lim supto0 Iu(x, t) I is finite-valued except possibly for a countable set E; (iii) lim inft~0 u(x, t) =u*(x) is in L1 on every compact subset of (-oo, oo); (iv) there exist positive constants K and /3 such that I U(x) I < K exp [fX2] where U(x)= S' f u*(z) dz dy; (v) lim inft,0 tZ2u(x, t)=O for x in E. Then