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The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions. (English) Zbl 0538.35048

The author examines the parabolic initial value problem \((\alpha)\quad u_ t=u_{xx},\quad 0<x<L,\quad t>0;\quad u(0,t)=u(x,0)=0,\quad u_ x(L,t)=\phi(u(L,t))\) where \(\phi(-\infty,1)\to(0,\infty)\) is continuously differentiable, monotone increasing and \(\lim_{u\to 1^- }\phi(u)=+\infty.\) He proves that there is a positive number \(L_ 0\) such that if \(L\leq L_ 0\), \(u(x,t)\leq 1-\delta\) for some \(\delta>0\) for all \(t>0\), while if \(L>L_ 0\), u(L,t) reaches one in finite time while \(u_ t(L,t)\) becomes unbounded in that time. The hyperbolic problem \((\beta)\quad u_{tt}=u_{xx}, 0<x<L\), \(t>0\); \(u(0,t)=u(x,0)=u_ t(x,0)=0,\quad u_ x(L,t)=\phi(u(L,t))\) is analyzed in the same manner.
Reviewer: K.Hawlitschek

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B35 Stability in context of PDEs
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