QUENCHING BEHAVIOR OF SEMILINEAR HEAT EQUATIONS WITH SINGULAR BOUNDARY CONDITIONS

In this article, we study the quenching behavior of solution to the semilinear heat equation v(t) = v(xx) + f (v), with f(v) = -v(-r) or (1 - v)(-r) and v(x)(0,t) = v(-P)(0,t), v(x)(a,t) = (1-v(a,t))(-q). For this, we utilize the quenching problem u(t) = u(xx) with u(x) (0, t) = u(-P)(0,t), u(x)(a,t) = (1 - u(a,t))(-q). In the second problem, if u(0) is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x = 0 (x = a) and u(t) blows up at quenching time. Further, we obtain a local solution by using positive steady state. In the first problem, we first obtain a local solution by using monotone iterations. Finally, for f(v) = -v(-r) ((1 - v)(-r)), if v(0) is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x = 0 (x = a) and v(t) blows up at quenching time.