The paper deals with solutions in $\bbfR^N$ to nonlinear parabolic equations of the form $u_t+L(u)+h(x,t,u)=0$, with initial condition $u(x,0)=u_0(x)$. $L(u)$ is an elliptic differential operator in divergence form with some structure conditions, which include the standard $p$-Laplacian operator, and $h(x,t,u)$ is a function which grows uniformly with $u$ at a sufficient rate. If $L(u)$ is the $p$-Laplacian then the rate of growth of $h$ is greather than $p-1$. The function $u_0(x)$ is assumed to be locally integrable in $\bbfR^N$, without any control of its growth at infinite. First, the authors establish a priori estimates of local type for a sequence of suitable approximate problems, then, passing to the limit and using the conditions in above, they prove existence of a solution for the initial problem. The corresponding results for elliptic problems were already studied. Also the regularity of the solution is investigated. In particular, for large rates of growth of $h$ with respect to $u$, the regularity of $u$ and $Du$ is improved with respect to the results known from the standard theory. Finally, the question of uniqueness of local solutions is discussed.