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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.

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