Asymptotic estimates to quenching solutions of heat equations with weighted absorptions

Article type: Research Article

Authors: Wang, Wei | Zheng, Sining

Affiliations: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China

Note: [] Corresponding author. E-mail: snzheng@dlut.edu.cn

Abstract: 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=(max x(f(x)/ϕp+1(x)))−1 as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/ϕp+1 for large M.

Keywords: heat equations, quenching time, quenching set, asymptotic estimates, local energy estimates, weighted absorptions

DOI: 10.3233/ASY-2010-1006

Journal: Asymptotic Analysis, vol. 70, no. 3-4, pp. 125-139, 2010