Propagation of strong singularities in semilinear parabolic equations with degenerate absorption

We study equations of the form (*) u(t) - Delta u + h(x)vertical bar u vertical bar(q - 1)u = 0 in a half space R-+(N+1). Here q > 1 and h is a continuous function in R-N, vanishing at the origin and positive elsewhere. Let (h) over bar (s) = e(-omega(s)/s2) and assume that omega(s)/s(2) is monotone on (0, 1) and tends to infinity as s -> 0. We show that, if omega satisfies the Dini condition and h(x) >= (h) over bar(vertical bar x vertical bar) then there exists a maximal solution of (*). This solution tends to infinity as t -> 0. On the contrary, if the Dini condition in the half space fails and h(x) <= (h) over bar (x), we construct a sequence of solutions whose initial data shrinks to the Dirac measure with infinite mass at the origin, but the limit of the sequence blows up everywhere on the positive time axis.