Short time asymptotic expansions of solutions of parabolic equations

Their procedure is a direct analog of the ray methods used by Keller [2] and others to treat elliptic equations. However, their expansions are formal, and we wish to show that they are indeed asymptotic to exact solutions. We shall do so for the case of one space dimension with U(X) independent oft and with the extra term q(x) u included on the left side of (1.1). In this case, the parameter h can be eliminated by introducing t’ = t/A. Then t’ --+ 0 as h --f 00, so we shall determine the asymptotic expansion of the solution for t’ small. We shall also set U,,(X) = 6(x 5) so that u is the Green’s function or fundamental solution of the initial value problem. In Section 2 we formulate the problem and obtain the formal asymptotic expansion of U. In Section 3 we show how the expansion can be used to find