Existence and non-existence results for parabolic systems with an Hardy-Leray potential

Abstract

In this paper we study the problem of existence or non existence of positive supersolution to the system$\{\begin{array}{cc}{u}_t-\Delta u={\lambda }_1\frac{u}{|x{|}^2}+f(v,\nabla v),& \text{ in }{\Omega }_T,\\ v_t-\Delta v={\lambda }_2\frac{v}{|x{|}^2}+g(u,\nabla u),& \text{ in }{\Omega }_T,\end{array}$ where $\Omega \subset R^N,N\ge 3$, is a regular domain containing the origin and:

i) $f=v^p,g=u^q$, $ii)$ $f=|\nabla v{|}^p,g=|\nabla u{|}^q$, $iii)$ $f=v^p,g=|\nabla u{|}^q$.

According to the form of the nonlinearities, we are able to get the existence of critical curves separating the existence and the non existence regions. In the case $f=v^p$ and $g=u^q$, we study the Cauchy system in $R^N\times (0,T)$. The existence of a Fujita type exponent is deeply analyzed.