ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF THE CAUCHY PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS

This paper is concerned with the Cauchy problem for the nonlinear parabolic equation tu = 1 u + F(x, t, u,. u) in R N x (0,8), u(x, 0) =.(x) in R N, where N = 1, F. C(R N x (0,8) x R x R N),.. L 8(R N) n L 1 (R N, (1 + | x| K) dx) for some K >= 0. We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t -> a and obtain the higher order asymptotic expansions of the solution in W (1,q) (R (N) ) with 1 a parts per thousand currency sign q a parts per thousand currency sign a infinity.