An integral equation for a Stefan problem with many phases and a singular source (Q2767713)

The authors' conclusion: ``This paper discusses the self-similar solutions \(\theta(x,t)= \theta(\eta)=\theta\left(x/\sqrt t\right)\) of the problem NEWLINE\[NEWLINE E(\theta)_t-A(\theta)_{xx}=\tfrac 1 t B(\theta); \quad \theta(0,t)=C>0,\;t>0; \;E\left( \theta(x,0) \right)=0. NEWLINE\]NEWLINE \noindent Here \(E(\theta)\) represents energy per unit volume at level (temperature) \(\theta\), \(A'(\theta)\) is the thermal conductivity and \(B(\eta)/t\) represents a singular source or sink depending of the sign of the function \(B\). It is assumed that \(E\) and \(A\) are monotone increasing functions, \(A\) being continuous, with \(E(0)=A(0)=0 \) and \(\lambda=E(0^+)>0\). The authors obtain for the inverse function \(\eta=\eta(\theta)\) an integral equation equivalent to the above problem and prove that under a certain hypothese on data there exists at least a solution of the corresponding integral equation''.