Life-span of solutions for a nonlinear parabolic system

In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system \(\partial _t u=\Delta u+|v|^{p-1} v,\; \partial _t v=\Delta v+|u|^{q-1} u,\) ${\partial }_{t}u=\mathrm{\Delta }u+{|v|}^{p-1}v,{\partial }_{t}v=\mathrm{\Delta }v+{|u|}^{q-1}u,$ \(q\ge p\ge 1,\; q>1\) $q\ge p\ge 1,q>1$ with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.