Influence of the Hardy potential in a semilinear heat equation

Abstract

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem where Ω⊂ℝ N , N ≥3, is a bounded regular domain such that 0∈Ω, p >1, and u 0 ≥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=ℝ N , 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).