FINDING THE CRITICAL DOMAIN OF MULTI-DIMENSIONAL QUENCHING PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS

摘要

Let Ω be a disc in R~2 with the center (0,0) and radius a, ∂Ω and Ω be its boundary and closure, respectively. Suppose that u is a function of τ, χ, and Ç. Further, assume that β is a positive number. In this paper, we investigate the multi-dimensional parabolic quenching problems with the second initial-boundary condition: ∂u/∂τ=∂~2u/∂χ~2+∂~2u/∂Ç~2+1/1-u for(χ,Ç,τ)∈Ω×(0,∞), u(χ,Ç,0)=uο(χ,Ç) for (χ,Ç)∈Ω, ∂u(χ,Ç,τ)/∂n=-β/α for τ＞0 and (χ,Ç)∈∂Ω where uο∈C~2(Ω) and uο (χ,Ç) ＜1 for (χ,Ç) ∈Ω is the outward normal derivative of u. We shall determine an approximated critical domain of some uο(χ,Ç) of the above problem by using a numerical method.