A parabolic nonlinear elliptic P.D.E. with singular potential

  • Wahid Sayeb Departement de Math\'ematiques, Facult\'edes sciences de Tunis,\\ Universit\'e Tunis El Manar, Campus Universitaire 2092, Tunis, Tunisia

Abstract

In this paper, we prove the existence and the non existence of nonnegative solutions of the quasilinear parabolic elliptic equation
\begin{displaymath}(P_{t,s,m})\left\{
\begin{array}{lll}
u_{t}-\Delta_{p}u-c\frac{u^{p-1}}{|x|^{p}}=u^{m}+sf &\;\;\textrm{ in }\;\Omega\times(0,t)\\
u(x,t)=0 &\;\;\textrm{ in }\;\partial\Omega\times(0,t)\\
u(x,0)=u_{0}(x)&\textrm{ in }\;\Omega,\\
\end{array} \right.
\end{displaymath}
where $N\geq3, \;\;1 < p < N,\;\; m > p-1,\;\; s\geq0,\;\;0<c< c_{N,p}=(\frac{N-p}{p})^{p},\;\;\Omega$ is a bounded regular domain containing the origin, and $u_{0} \geq 0$, $f \geq 0$ measurable function with suitable hypotheses. In the case of existence result ($1<m<m_{0}$), there exists $s_{0}$ such that for $0\leq s\leq s_{0}, (P_{t,s,m})$ has a weak solutions, and blow-up at finite time. In the case of nonexistence result ($m>m_{0}$), we analyze blow-up phenomena for approximated problems in connection with a comparison principle.

Published
2021-05-23