Bounded solutions in anisotropic degenerate parabolic problems with a singular term

In this paper, we study the existence of bounded solutions for a nonlinear anisotropic parabolic equation with degenerate coercivity and a singular term on the right-hand side. The model problem considered is as follows {{partial derivative u/partial derivative t-& sum;D-N(i=1)i(u(i)(p)-1(1+Du)(-1)Du+|Du|(p)(i)-2Du/(1+|u|)(theta))=f/u(gamma)in Q, u(x,0)=0 on Omega, u=0 on Gamma, where Omega is a bounded open subset of R-N N >= 2, T>0, 2 <= p(i)<N for every i=1,& mldr;,N, theta,gamma >= 0, 0 <= f is an element of L-m(Q) with m>N/p+1 (p((sic)) defined in (2.1)) and Q=Omega x(0,T). The main idea in the proof is based on Stampacchia's lemma, which allows us to obtain a priori estimates by making a suitable choice of a test function.