Abstract
We consider the differential inequality with a nonlinear memory ∂tu−Δu≥1Γ(σ)∫0t(t−s)σ−1|u(s,x)|pds+w(x),t>0,x∈R+N, where N≥2, p>1, σ>0, and w⁄≡0. Under certain conditions on the initial value and the inhomogeneous term w, we show that the Fujita critical exponent is given by pσ∗=∞, for all σ>0. Next, we consider the limit case of the above problem as σ→0+, i.e. ∂tu−Δu≥|u|p+w(x),t>0,x∈R+N. A discontinuity phenomenon of the critical exponent is observed. Namely, we show that the Fujita critical exponent for the limit problem is equal to N+1N−1, which is different to limσ→0+pσ∗.