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

Abstract

In this article, we study the following noncoercive quasilinear parabolic problem ∂u∂t−diva(x,t,u,∇u)+ν∣u∣s−1u=λ∣u∣p−2u∣x∣p+finQT,u=0onΣT,u(x,0)=u0inΩ,\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∈L1(QT)f\in {L}^{1}\left({Q}_{T}) and u0∈L1(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.

Keywords

• quasilinear parabolic equations
• noncoercive equations
• hardy potential
• entropy solutions
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