Asymptotic estimates to quenching solutions of heat equations with weighted absorptions

摘要

This paper studies heat equations with weighted nonlinear absorptions of the form ut = uxx - M f(x)u~(-p) in (- 1,1) x (0, T) subject to Dirichlet boundary conditions u(-1,t) = u(1,t) = 1 and initial dataφ(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/max_x(f(x)/φ~(p+1)(x)) quenching set concentrates near the maximum points of f/φ~(p+1) for large M.