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 derivative(t)u = Delta u + |v|(p-1)v, partial derivative(t)v=Delta v + |u|(q-1)u, q >= p >= 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.