Boundedness and Regularity of Solutions for Nonlinear Parabolic Problem With Two Singularities

Abstract

 We examined the boundedness and regularity of weak solutions for a type of nonlinear parabolic equations with singular natural growth gradient terms. The equations also have a singular right-hand side with $f\in L^{m}(Q)$, ($m\geq 1$). With appropriate test functions, we can use Stampacchia's Lemma to demonstrate that the solutions $u(x, t)$ are bounded when $m>\frac{N}{p}+1$. Additionally, we can establish a regularity result if $f\in L^{1}(Q_T)$.