Remarks on Quenching 201 Quenching Occurs
نویسندگان
چکیده
Consider the parabolic problem u t div (a(u;ru)ru) = u p (1) for t > 0, x 2 R n under initial and boundary conditions u = 1, say. Since p is assumed positive, the right hand side becomes singular as u ! 0. When u reaches zero in nite or in nite time, one says that the solution quenches in nite or in nite time. This article gives a survey of results on this kind of problem and emphasizes those that have been obtained at the SFB 123 in Heidelberg. It is an updated version of an invited survey lecture at the International Congress of Nonlinear Analysts in Tampa, August 1992. To be speci c, I shall cover existence and nonexistence of quenching points, asymptotic behaviour of the solutions in space and time near the quenching points, qualitative behaviour, application to mean curvature ow and phase transitions, reaction in porous medium ow etc.. The tools are variational methods and suitable maximum principles. Many of the results presented in this article were obtained with my coauthors Acker, Dziuk, Fila, Kersner and Levine, but related results will also be mentioned. 1991 Mathematics Subject Classi cation: 35K65, 35K57, 35K60, 35B05, 35B65
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