SINGULAR PARABOLIC PROBLEMS WITH POSSIBLY CHANGING SIGN DATA

We show the existence of bounded solutions u epsilon L-2(0, T; H-0(1)(Omega)) for a class of parabolic equations having a lower order term b(x, t, u, del u) growing quadratically in the del u-variable and singular in the u-variable on the set {u= 0}. We refer to the model problem {u(t)-Delta u=b(x,t)vertical bar del u vertical bar(2) / vertical bar u vertical bar(k) +f(x,t) in Omega x (0,T) {u(x,t) =o on partial derivative Omega x(0,T) {u(x,0) -u(0)(x) in Omega where Omega is a bounded open subset of R-N, N >= 2, 0 < T <+infinity and 0< k < 1. The data f(x, t), uo(x) can change their sign, so that the possible solution u can vanish inside QT = Omega x(0, T) even in a set of positive measure. Therefore, we have to carefully define the meaning of solution. Also b(x, t) can have a quite general sign.