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 $g(0) = +\infty$ andor $h(0)= +\infty$, 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.