Finite-time blow-up for inhomogeneous parabolic equations with nonlinear memory

In this paper, we study the effect of an inhomogeneity w = w(x) on the finite-time blow-up of solutions to the nonlinear heat equation partial derivative(t)u - Delta u = 1/Gamma(1 - gamma) integral(t)(0) (t - s)(-gamma) vertical bar u(s)vertical bar(p) ds + w(x), (t, x) is an element of (0, T) x R-N, where N >= 1, 0 < gamma < 1 and p > 1. It is well known that in the homogeneous case w equivalent to 0, the Fujita critical exponent is given by p(*) = max {1/gamma, 1 + 4 - 2 gamma/(N - 2 + 2 gamma)(+)}. In the case integral(RN) w(x) dx > 0 and u(0, center dot) >= 0, we prove that the critical exponent is equal to 8, which means that for all p > 1, we have a finite-time blow-up.