FAST DIFFUSION NONLINEAR HEAT-EQUATIONS AND A RELATED SINGULAR ELLIPTIC PROBLEM

We provide necessary conditions on u(0) for the solvability of the pro blem. partial derivative u/partial derivative t = div (u(m-1)del u) in R(N) X (0,T), u(x,0) = u(0)(x) in R(N), where T > 0, m < min{0,(N - 2 )/N} or m = 0, N = 2. We also find both, necessary and sufficient cond itions for the solvability of the elliptic problem Delta u + u(-nu) = f(x), x is an element of R(N), where nu > 0, N greater than or equal t o 2, which is obtained by discretization of the parabolic problem when nu = -1/m.