A Stefan problem in combustion
The authors consider a special model in the theory of combustion which has been introduced by Margolis, Matkowsky, Sivashinsky, Kaper, and Leaf and leads to a Stefan problem for the determination of the temperature T and the concentration of the combustible reactant y: \[ \partial_ ty=- y(1+(\alpha -1)H(T-T_ m))\exp (-E/RT), \] \[ \partial_ t(T+\gamma yH(T-T_ m))=\lambda \Delta T+\beta y(1+(\alpha -1))H(T-T_ m)\exp (- E/RT) \] in \(\Omega\) \(\times [0,\tau)\) (\(\Omega\) : some threedimensional cylinder), \(y(x,0)=y_ 0(x)\), \(y_ 0\in L_{\infty}(\Omega)\), \(y_ 0\geq 0\), \(T(x,0)=T_ 0(x)\), \(T_ 0\in W^{1,2}(\Omega)\), \(t_ 0\geq \delta >0\), \((\partial /\partial n)T=0\) on \(\partial \Omega \times [0,\tau)\) (n: normal to \(\partial \Omega)\). Here \(\alpha\),\(\beta\),... denote several parameters. R, E and \(T_ m\) denote a gas constant, the (large) activation energy and the melting temperature. The authors prove the existence of a weak solution (y,T) of this problem and give a characterization of the phase change set \(\{T=T_ m\}\). From this characterization they are able to conclude that in physically interesting cases (i.e. if \(T_ 0<T_ m\) and \(T(t_ 0)>T_ m\) for some \(t_ 0>0)\) the phase change set is not a front but a mushy region.
scientific article; zbMATH DE number 4124280
scientific article