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FINDING THE CRITICAL DOMAIN OF MULTI-DIMENSIONAL QUENCHING PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS | Neural, Parallel, and Scientific Computations

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

 Title FINDING THE CRITICAL DOMAIN OF MULTI-DIMENSIONAL QUENCHING PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS Publication Type Journal Article Year of Publication 2017 Authors CHAN W.Y., LIU H.T. Journal Neural, Parallel, and Scientific Computations Volume 25 Start Page 19 Pagination 9 Date Published 2017 ISSN 1061-5369 Keywords 35J47, 35J60, 35K20, 35K55 Abstract 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. URL https://acadsol.eu/npsc/articles/25/1/2.pdf Refereed Designation Refereed