A limiting regularity result for some parabolic problems with data in Zygmund spaces

The aim of this paper is to study the model problem:{partial derivative u/partial derivative t - Delta(p)u = f in Qu = 0 on partial derivative Omega x (0, T)u(x, 0) = u(0) in Omega.The main purpose of this work is to prove the existence and a limiting regularity result for the solution u of the above problem having right-hand side f is an element of L-beta (0, T; LLog(alpha) L(Omega)). In particular, we will consider the cases:(i) If p >= 2, beta = 1, and alpha > N-1/N, then u is an element of Lp-1(0, T, W-0(1,(q) over bar) (Omega)) ((q) over bar = N(p-1)/N-1).(ii) If 2 - 1/N < p < N, beta = N', and alpha = N-1/N, then u is an element of L (q) over bar (0, T, W-0(1,(q) over bar) (Omega)).(iii) If p = N, beta = N', and alpha = N-1/N, then u is an element of L-N(0, T, W-0(1,N) (Omega)).(iv) If p >= 2, f is an element of L-1(0, T; L(m)Log(alpha) L(Omega)), and alpha > N-m/N, then u is an element of Lp-1(0, T, W-0(1,m*(p-1)) (Omega)).