*preprint*

**Inserted:** 19 jan 2024

**Year:** 2021

**Abstract:**

In this paper we prove existence of nonnegative solutions to parabolic
Cauchy-Dirichlet problems with superlinear gradient terms which are possibly
singular. The model equation is \[
u_t - \Delta_pu=g(u)

\nabla u

^q+h(u)f(t,x)\qquad \text{in
}(0,T)\times\Omega, \] where $\Omega$ is an open bounded subset of
$\mathbb{R}^N$ with $N>2$, $0<T<+\infty$, $1<p<N$, and $q<p$ is superlinear.
The functions $g,\,h$ are continuous and possibly satisfying $g(0) = +\infty$
and*or $h(0)= +\infty$, with different rates. Finally, $f$ is nonnegative and
it belongs to a suitable Lebesgue space. We investigate the relation among the
superlinear threshold of $q$, the regularity of the initial datum and the
forcing term, and the decay rates of $g,\,h$ at infinity.
*