Let ${{\textohm}}$ be a disc in ${ R^2}$\ with the center (0, 0) and radius a, ${ ∂{\textohm} }$ and ${ \bar{{\textohm}} }$\ 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: $$ { \frac{∂u}{∂τ} = \frac{∂^2u}{∂χ^2}\ + \frac{∂^2u}{∂ζ^2} +\frac{1}{1 - u} \ for\ \ (χ, ζ, τ) ∈ {\textohm} {\texttimes} (0, $\infty$), }$$ $${\ u (χ, ζ, 0) = u_0 (χ, ζ) \ for \ (χ, ζ) ∈ \bar{{\textohm}}, \frac{∂u (χ, ζ, τ)}{∂n} = - \frac{β}{a} \ for \ τ \> 0 \ and \ (χ, ζ) ∈ ∂{\textohm}, }$$\ where ${ u_0 ∈ C^2 \left( \bar{{\textohm}} \right)}$\ and ${ u_0 (χ, ζ) \ \< \ 1}$ \ for ${\ (χ, ζ) ∈ \bar{{\textohm}}, }$ 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.

}, keywords = {35J47, 35J60, 35K20, 35K55}, issn = {1061-5369}, url = {https://acadsol.eu/npsc/articles/25/1/2.pdf}, author = {W.Y. CHAN and H.T. LIU} }