Quasi-linear parabolic equations having a superlinear gradient term which depends on the solution

TL;DR

This paper investigates the existence and regularity of solutions to quasi-linear parabolic equations with superlinear gradient terms, considering different initial data spaces and extending previous results in the field.

Contribution

It introduces new existence and regularity results for such equations with initial data in Orlicz and Lebesgue spaces, broadening the understanding of these complex PDEs.

Findings

Existence of solutions with initial data in Orlicz spaces.

Regularity results under smallness conditions in Lebesgue spaces.

Extension of previous frameworks to more general data spaces.

Abstract

In this paper, we study existence and regularity for solutions to parabolic equations having a superlinear lower order term depending on both the solution and its gradient. Two different situations are analyzed. On the one hand, we assume that the initial datum belongs to an Orlicz space of exponentially summable functions. On the other, data in an appropriate Lebesgue space satisfying a smallness condition are considered. Our results are coherent with those of previous papers in similar frameworks.