Chaos expansion of heat equations with white noise potentials

The asymptotic behavior as t --> infinity of the solution to the following stochastic heat equations [GRAPHICS] is investigated, where w is a space-time white noise or a space white noise. The use of lozenge means that the stochastic integral of 10 (Skorohod) type is considered. When d = 1, the exact L-2 Lyapunov exponents of the solution are studied. When the noise is space white and when d < 4 it is shown that the solution is in some "flat" L-2 distribution spaces. The Lyapunov exponents of the solution in these spaces are also estimated. The exact rate of convergence of the solution by its first finite chaos terms are also obtained