On the Boundary Value of a Solution of the Heat Equation

Let U be an open subset of R", and let P(x, Dx) be a linear differential operator with analytic coefficients defined on a neighborhood of the closure [£/] of U. Suppose that the boundary dU of U is smooth (i.e., non-singular and analytic) and that dU is non-characteristic with respect to P at each point in dU. Then it is well-known ([7], [11]) that the boundary value of a hyperfunction solution of the equation Pu = 0 on U is a well-defined hyperfunction. However, little is known about the characterization of a solution whose boundary value determines a hyperfunction near a characteristic boundary point. The purpose of this article is to discuss this problem for one n-l special case, i.e., the pair of the heat operator d/dt — A = d/dt — J^ d/dxj def j=i and the domain {(£, x)eR"; t > 0}. Our main result (Theorem 1 below) asserts that, (i) if a C°°-solution u(t, x) does not behave too wildly as t|0, and (ii) if u(t, x) uniformly tends to zero outside a compact set K a R"' as *iO, then we can assign a compactly supported hyperfunction g(x) to u(t, x) so that the vanishing of g(x) entails the vanishing of u(t, x) itself. Furthermore we can find such a tame solution u(t, x) of the heat equation for any compactly supported hyperfunction g(x). [See Theorem 1 for the precise statement. Note also that a hyperfunction supported by a compact set, say L, is an analytic functional with the real carrier L.] Let us note the following two facts: First, if u(t, x) tends to infinity too rapidly as t j 0, then our procedure will not assign a hyperfunction g(x). (Cf. §2(i)) Second, we know (see [3], for example) that there exists a hyperfunction e(t, x) (xeR) supported by {(t, x)eR; t = 0, x ^ 0} satisfying the equation