Asymptotic estimates to quenching solutions of heat equations with weighted absorptions

This paper studies heat equations with weighted nonlinear absorptions of the form u_t=u_xx−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=(max _x(f(x)/ϕ^p+1(x)))⁻¹ as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/ϕ^p+1 for large M.