Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}}^N$ with absorbing first order terms, whose model is ${ {l} u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{in}, (0,T) \times $\mathbb{R}$^N,, \[1.5 ex] u(0,x)= u_0 (x) &box{in}, $\mathbb{R}$^N.$where$T>0$,$N\geq 2$,$1<q\leq 2$,$f(t,x)\in L^1( 0,T; L^1_{\rm loc} ($\mathbb{R}$^N))$and$u_0\in L^1_{\rm loc} ($\mathbb{R}$^N)$.