Uniqueness in the Cauchy problem for parabolic equations

Abstract:

In a classical paper S. Täcklind (Nova Acta Soc. Sci. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind’s proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green’s function for a half cylinder. G. N. Zolotarev (Izv. Vysš. Učebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind’s uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green’s function. We have now shown, using a new approach which replaces the construction based upon a Green’s function by an appropriate comparison solution of the maximizing equation (C. Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind’s condition is sufficient without any regularity conditions on the coefficients.