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

Let ${Ω}$ be a disc in ${ R^2}$  with the center (0, 0) and radius a, ${ ∂Ω }$ and ${ \bar{Ω} }$  be its boundary and closure, respectively.

Suppose that ${u}$ is a function of ${ τ, χ,}$ and ${ ζ.}$ 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: $$ { \frac{∂u}{∂τ} = \frac{∂^2u}{∂χ^2} + \frac{∂^2u}{∂ζ^2} +\frac{1}{1 − u} \  for  \ (χ, ζ, τ) ∈ Ω × (0, ∞), }$$ $${ u (χ, ζ, 0) = u_0 (χ, ζ) \ for \ (χ, ζ) ∈ \bar{Ω}, \frac{∂u (χ, ζ, τ)}{∂n} = − \frac{β}{a} \ for \ τ > 0 \ and \ (χ, ζ) ∈ ∂Ω, }$$ where ${ u_0 ∈ C^2 \left( \bar{Ω} \right)}$ and ${ u_0 (χ, ζ) \ < \ 1}$  for ${ (χ, ζ) ∈ \bar{Ω}, }$ and ${ ∂u/∂n }$ is the outward normal derivative of ${ u }$. We shall determine an approximated critical domain of some ${ u_0 (χ, ζ) }$ of the above problem by using a numerical method.