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Remarks on quenching. (English) Zbl 0866.35057

This paper summarizes results obtained by the author and others on equations similar to \(u_t-\Delta u= -u^{-p}\). In one dimension, the Laplacian may be replaced by operators of the form \((\varphi (u_x)_x)\) or \((u^m)_{xx}\). When the solution of such an equation reaches zero, some derivative of the solution must become infinite; this behavior is known as quenching.
The author considers several elements of this quenching behavior. The first is a set of sufficient conditions for solutions to quench in finite time; the basic set of conditions can be easily summarized: there are no stationary solutions to the problem. Next, he discusses the asymptotic behavior of a solution near a quenching point. After a brief examination of the location of quenching points, he finishes by asking what kind of existence theory (for a suitable weak solution) is available after the quenching time. The discussion here is quite informal, but it is useful to read the work of one of the primary researchers on the quenching problem. A key part of this paper is its examination of the quenching problem in context; unlike many papers, where quenching is studied for its own sake, this one talks about its connections to blow-up problems and about what kinds of results are possible or likely for the solutions.

MSC:

35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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