This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem

where Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.

There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.

The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).