Quasilinear class of noncoercive parabolic problems with Hardy potential and L1-data

Abstract

Abstract In this article, we study the following noncoercive quasilinear parabolic problem ∂ u ∂ t − div a ( x , t , u , ∇ u ) + ν ∣ u ∣ s − 1 u = λ ∣ u ∣ p − 2 u ∣ x ∣ p + f in Q T , u = 0 on Σ T , u ( x , 0 ) = u 0 in Ω , \left\{\begin{array}{ll}\frac{\partial u}{\partial t}-\hspace{0.1em}\text{div}\hspace{0.1em}a\left(x,t,u,\nabla u)+\nu {| u| }^{s-1}u=\lambda \frac{{| u| }^{p-2}u}{{| x| }^{p}}+f& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{Q}_{T},\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\Sigma }_{T},\\ u\left(x,0)={u}_{0}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. with f ∈ L 1 ( Q T ) f\in {L}^{1}\left({Q}_{T}) and u 0 ∈ L 1 ( Ω ) {u}_{0}\in {L}^{1}\left(\Omega ) and show the existence of entropy solutions for this noncoercive parabolic problem with Hardy potential and L1-data.