Higher regularity of solutions of singular parabolic equations with variable nonlinearity

We study the global regularity of solutions of the homogeneous Dirichlet problem for the parabolic equation with variable nonlinearity u(t) - div (vertical bar del u vertical bar(p(x, t)-2)del u - f) = 0 in Q(T) = Omega x (0, T), where p(x, t), f(x, t) are given functions of their arguments, n >= 2 and 2n/n+2 < p(x, t) <= 2. Conditions on the data are found that guarantee the existence of a unique strong solution such that u(t) is an element of L-2(QT) and vertical bar del u vertical bar is an element of L-infinity (0, T; L-p(center dot) (Omega)). It is shown that if partial derivative Omega is an element of C1+beta with beta is an element of (0, 1), p and f are Holder-continuous in <(Omega)over bar> x (0, T], D(i)f(j) is an element of L-2(Q(T)) and vertical bar del p vertical bar is an element of L-infinity (Q(T)), then for every strong solution D(ij)(2)u is an element of L-2(Omega x (s, T)) with any s is an element of (0, T).