Some new results on a Stefan problem in a concentrated capacity
Let \(\Omega\subset\mathbb{R}^ n\), \(n\geq 2\), be a bounded regular set and let \(\theta=\beta(u)\) represent the constitutive equation relating temperature \((\theta)\) with enthalphy \((u)\). The following system \[ \partial\theta/\partial t-\Delta\theta=\varphi \quad\text{in}\quad \Omega\times(0,T), \quad \partial u/\partial t-\Delta_ g\theta- \partial\theta/\partial u=f, \quad \theta=\beta(u), \quad \partial\Omega\times(0,T)=\beta \] with initial data \(\theta=\theta_ 0\) in \(\Omega\) and \(u=u_ 0\) on \(\Gamma\) and \(f\), \(\varphi\) given function, \(\Delta_ g\) the Laplace-Beltrami operator, and \(\nu\) the inward normal to \(\Gamma\), is a scheme for the description of a phase change process in a concentrated capacity. The problem with \(u_ 0=0\), \(\theta_ 0=0\), \(\varphi=0\) has been solved by the same author in a previous paper. Here an existence and uniqueness theorem is obtained for the general case. The extension is not trivial. The technique consists of solving a family of regularized problems, introducing smooth approximations \(\beta_ \varepsilon\) of \(\beta\) and then in obtaining some uniform estimates for the regularized solutions in a suitable space. A compactness argument provides existence of a weak solution defined in some natural way. Uniqueness is proved by the same method used for showing that the regularized problems have only one solution.
scientific article; zbMATH DE number 69231