Nonexistence thresholds for semilinear heat inequalities with an inner-boundary inverse-square Hardy potential on an annulus

We study a semilinear heat inequality on an annulus with inner boundary Σ 1 and outer boundary Σ 2 . The problem involves the inverse-square Hardy potential 𝜆 / 𝛿 1 ⁢ ( 𝑥 ) 2 , where 𝛿 1 ⁡ ( 𝑥 ) = d i s t ⁡ ( 𝑥 , Σ 1 ) and 𝜆 ∈ ℝ , and the weighted power nonlinearity 𝛿 1 ⁢ ( 𝑥 ) − 𝛾 ⁢ | 𝑢 | 𝑝 , where 𝛾 ∈ ℝ and 𝑝 > 1 . A nonhomogeneous Dirichlet-type boundary condition is imposed on Σ 2 . The boundary datum 𝑓 = 𝑓 ⁡ ( 𝑥 ) belongs to 𝐿 1 ⁡ ( Σ 2 ) , is nontrivial, and satisfies ∫ Σ 2 𝑓 ⁡ 𝑑 ⁢ 𝑆 ≥ 0 . We provide a sharp description of the existence and nonexistence regimes in terms of λ , γ , and p . In particular, when 𝜆 ≥ − 1 / 4 and 𝛾 > 2 , we identify a sharp Fujita-type critical exponent 𝑝 𝑐 ⁡ ( 𝜆 , 𝛾 ) . As a consequence, we also obtain the corresponding sharp nonexistence result for the associated elliptic inequality.