The paper deals with solutions in

${\mathbb{R}}^{N}$ to nonlinear parabolic equations of the form

${u}_{t}+L\left(u\right)+h(x,t,u)=0$, with initial condition

$u(x,0)={u}_{0}\left(x\right)$.

$L\left(u\right)$ 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

$L\left(u\right)$ is the

$p$-Laplacian then the rate of growth of

$h$ is greather than

$p-1$. The function

${u}_{0}\left(x\right)$ is assumed to be locally integrable in

${\mathbb{R}}^{N}$, 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.