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