Asymptotic estimates to quenching solutions of heat equations with weighted absorptions

This paper studies heat equations with weighted nonlinear absorptions of the form ut = uxx − Mf (x)u −p in (−1, 1) × (0, T ) subject to Dirichlet boundary conditions u(−1, t) = u(1, t) = 1 and initial data φ(x). The asymptotic estimates to quenching time and set of solutions as M → +∞ is established by local energy estimates. It is obtained that the quenching time T ∼ m p+1 · M −1 with m = 1 maxx(f (x)/φp+1(x)) as M → +∞. It is shown also how the quenching set concentrates near the maximum points of f/ φ p+1 for large M.