Self-similar blow-up solutions for the supercritical parabolic Hardy-Hénon equation

We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-Hénon equation ∂ 𝑡 𝑢 = Δ ⁢ 𝑢 + | 𝑥 | 𝜎 ⁢ 𝑢 𝑝 , ( 𝑥 , 𝑡 ) ∈ ℝ 𝑁 × ( 0 , ∞ ) , in dimension 𝑁 ≥ 3 and the range of exponents 𝜎 ∈ ( − 2 , ∞ ) , 𝑝 > 𝑝 𝑆 ⁡ ( 𝜎 ) : = 𝑁 + 2 ⁢ 𝜎 + 2 𝑁 − 2 . We establish the existence of self-similar blow-up solutions for any 𝑝 > 𝑝 𝑆 ⁡ ( 𝜎 ) , provided 𝜎 ≥ 2 . Moreover, we prove that, if k is any natural number and 𝜎 ≥ 4 ⁢ 𝑘 − 2 , the parabolic Hardy-Hénon equation has at least k different self-similar blow-up solutions for any 𝑝 > 𝑝 𝑆 ⁡ ( 𝜎 ) . These results are in a stark contrast with the standard reaction-diffusion equation ∂ 𝑡 𝑢 = Δ ⁢ 𝑢 + 𝑢 𝑝 , ( 𝑥 , 𝑡 ) ∈ ℝ 𝑁 × ( 0 , ∞ ) , for which non-existence of any self-similar solution has been established, provided p overpasses the Lepin exponent 𝑝 𝐿 : = 1 + 6 𝑁 − 1 0 , 𝑁 ≥ 1 1 . For 𝜎 ∈ ( − 2 , 2 ) , we derive the expression of generalized Lepin exponents 𝑝 𝐿 ⁡ ( 𝜎 ) for 𝜎 ∈ ( 0 , 2 ) , respectively ̅ ̅ ̅ ̅ ̅ ̅ ̅ 𝑝 𝐿 ⁡ ( 𝜎 ) for 𝜎 ∈ ( − 2 , 0 ) , and prove existence of self-similar solutions with finite time blow-up for 𝑝 ∈ ( 𝑝 𝑆 ⁡ ( 𝜎 ) , 𝑝 𝐿 ⁡ ( 𝜎 ) ) , respectively 𝑝 ∈ ( 𝑝 𝑆 ⁡ ( 𝜎 ) , ̅ ̅ ̅ ̅ ̅ ̅ ̅ 𝑝 𝐿 ⁡ ( 𝜎 ) ) . Numerical evidence of the optimality of these exponents is also included.