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The phenomenon of quenching: a survey. (English) Zbl 0581.35037
Trends in the theory and practice of non-linear analysis, Proc. 6th Int. Conf., Arlington/Tex. 1984, North-Holland Math. Stud. 110, 275-286 (1985).
[For the entire collection see Zbl 0564.00006.]
The author surveys some recent results concerning the behavior of solutions of nonlinear evolutionary equations when the nonlinearity has a pole at a finite value of the solution. Simple examples of such equations are \(u_ t=u_{xx}+1/(1-u)\) or \(u_{tt}=u_{xx}+1/(1-u)\). Results for these, for higher dimensional problems and for related problems in which the nonlinearity occurs in the boundary condition are also discussed.

MSC:
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs