Degenerate diffusion and the Stefan problem. (English) Zbl 0562.35049
The authors consider the first initial-boundary-value problem for the equation \(u_ t=(\phi (u))_{xx}+f(x,u)\) (rather, its weak form) in which \(\phi\) (s) is constant on an interval (\(\alpha\),\(\beta)\) (finite or semiinfinite with \(\beta =\infty)\) and otherwise strictly increasing. In the context of the Stefan problem, this corresponds to seeking the enthalpy u of a one- or two-phase material when it is given initially (\(\phi\) (u) is its temperature), and there may be nonlinear internal heating (or cooling) according to the function f.
Although a great many qualitative properties of the solution are developed, special attention is focused on properties of the mushy region (where \(u\in [\alpha,\beta])\), if it is an interval at time \(t=0\) and the boundary conditions hold u below \(\alpha\) at the boundary. Then if \(f\leq 0\), it remains a monotonic decreasing interval and vanishes after a finite time. The same is true in the case of the 2-phase problem \((\beta <\infty)\) when one boundary value is less than \(\alpha\) and the other greater than \(\beta\). If \(f\geq 0\) and is large enough, on the other hand, then a mushy region will develop in time and persist, even if it is not present initially. Most of the results are obtained with the use of comparison principles.
Although a great many qualitative properties of the solution are developed, special attention is focused on properties of the mushy region (where \(u\in [\alpha,\beta])\), if it is an interval at time \(t=0\) and the boundary conditions hold u below \(\alpha\) at the boundary. Then if \(f\leq 0\), it remains a monotonic decreasing interval and vanishes after a finite time. The same is true in the case of the 2-phase problem \((\beta <\infty)\) when one boundary value is less than \(\alpha\) and the other greater than \(\beta\). If \(f\geq 0\) and is large enough, on the other hand, then a mushy region will develop in time and persist, even if it is not present initially. Most of the results are obtained with the use of comparison principles.
Reviewer: P.Fife
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35R35 | Free boundary problems for PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
Keywords:
degenerate diffusion; Stefan problem; qualitative properties; mushy region; comparison principlesReferences:
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