On some parabolic equations involving superlinear singular gradient terms

TL;DR

This paper establishes the existence of solutions for certain parabolic equations with superlinear and possibly singular gradient terms, analyzing how solution existence depends on the interplay of parameters, initial data, and forcing terms.

Contribution

It provides new existence results for parabolic equations with superlinear, potentially singular gradient terms, considering various growth conditions and data regularities.

Findings

Existence of nonnegative solutions under superlinear gradient conditions

Relation between superlinear threshold and data regularity

Impact of singularities in coefficients on solution existence

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.