*preprint*

**Inserted:** 19 jan 2024

**Year:** 2017

**Abstract:**

We study the existence of solutions of a nonlinear parabolic problem of
Cauchy-Dirichlet type having a lower order term which depends on the gradient.
The model we have in mind is the following: \[ \begin{cases}\begin{split} &
u_t-\text{div}(A(t,x)\nabla u

\nabla u

^{p-2})=\gamma

\nabla u

^q+f(t,x)
&\qquad\text{in } Q_T,\\ & u=0 &\qquad\text{on }(0,T)\times \partial \Omega,\\
& u(0,x)=u_0(x) &\qquad\text{in } \Omega, \end{split}\end{cases} \] where
$Q_T=(0,T)\times \Omega$, $\Omega$ is a bounded domain of $\mathrm{R}^N$, $N\ge
2$, $1<p<N$, the matrix $A(t,x)$ is coercive and with measurable bounded
coefficients, the r.h.s. growth rate satisfies the superlinearity condition \[
\max\left\{\frac{p}{2},\frac{p(N+1)-N}{N+2}\right\}<q<p \] and the initial
datum $u_0$ is an unbounded function belonging to a suitable Lebesgue space
$L^\sigma(\Omega)$. We point out that, once we have fixed $q$, there exists a
link between this growth rate and exponent $\sigma=\sigma(q,N,p)$ which allows
one to have (or not) an existence result. Moreover, the value of $q$ deeply
influences the notion of solution we can ask for. The sublinear growth case
with \[ 0<q\le\frac{p}{2} \] is dealt at the end of the paper for what concerns
small value of $p$, namely $1<p<2$.