ASYMPTOTIC-BEHAVIOR OF THE FUNDAMENTAL SOLUTION OF A 2ND-ORDER PARABOLIC EQUATION

We study the asymptotic as t --> infinity of the fundamental solution (FS) G(x,s,t) of the Cauchy problem for the parabolic equation G(t) - G(xx) + alpha(x)G = 0, x is an element of R(1), t > 0. We suppose that the coefficient alpha(x) can be written as x --> +/- infinity in the form alpha(x) = alpha(2)(+/-)x(-2) + phi(x), where the function phi(x) has an asymptotic expansion as x --> +/- infinity in positive powers of x(-1) and \phi(x)\ = o(\x\(-2)). We construct and justify the asymptotic expansion of the FS G(x,s,t) as t --> infinity up to any power of t(-1) for the whole plane x,s is an element of R(1).