The Quenching of Solutions of Linear Parabolic and Hyperbolic Equations with Nonlinear Boundary Conditions

In this paper we examine the initial-boundary value problems (a) limu__.-q,(u)= +o. For problem (a) we show that there is a positive number Lo such that if L<-L o, u(x,t)<_l-8 for some 8>0 for all t>0, while if L>Lo, u(L,t) reaches one in finite time while ut(L,t becomes unbounded in that time. For problem (fl) it is shown that if L is sufficiently small, then u(L, t)_<-8 for all t>0 while if L is sufficiently large and folq(r/) dr/< , u(L,t) reaches one in finite time whereas if fq(r/) dr/= o, u(L, t) reaches one in finite or infinite time. In either of the last two situations ut(L, t) becomes unbounded if the time interval is finite. If u reaches one in infinite time, then fd u(x, t) dx and u(x, t) are unbounded on the half line and half strip respectively.