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 $$left{begin{array}{l@{quad}l}u_t- Delta u +u |nabla u|^q = f(t,x) quad &{rm in}, (0,T) times mathbb{R}^N,,u(0,x)= u_0 (x) &{rm in}, mathbb{R}^N ,,quadend{array}right.$$ where ${T >0 , , Ngeq 2,, 1 < q leq 2,, f(t,x)in L^1left( 0,T; L^1_{rm loc} left(mathbb{R}^Nright)right)}$ and ${u_0in L^1_{rm loc}left(mathbb{R}^{N}right)}$ .