2025/11/04 by Iagar, Razvan Gabriel, Muñoz, Ana I., Sánchez, Ariel
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2511.02511
We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-Hénon equation ∂tu=Δu+|x|σup, (x,t)∈ℝN×(0,∞), in dimension N≥3 and the range of exponents σ∈(-2,∞), pgt;pS(σ):=(N+2σ+2)/(N-2). We establish the existence of self-similar blow-up solutions for any p>pS(σ), provided σ≥2. Moreover, we prove that, if k is any natural number and σ≥ 4k-2, the parabolic Hardy-Hénon equation has at least k different self-similar blow-up solutions for any p>pS(σ). These results are in a stark contrast with the standard reaction-diffusion equation ∂tu=Δu+up, (x,t)∈ℝN×(0,∞), for which non-existence of any self-similar solution has been established, provided p overpasses the Lepin exponent pL:=1+(6)/(N-10), N≥11. For σ∈(-2,2), we derive the expression of generalized Lepin exponents pL(σ) for σ∈(0,2), respectively pL(σ) for σ∈(-2,0), and prove existence of self-similar solutions with finite time blow-up for p∈(pS(σ),pL(σ)), respectively p∈(pS(σ),pL(σ)). Numerical evidence of the optimality of these exponents is also included.