On some parabolic equations involving superlinear singular gradient terms

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 \[ u_t - Δ_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text{in }(0,T)\timesΩ, \] 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.

References in corpus (1)

• Regularizing effects concerning elliptic equations with a superlinear gradient term