In this article, we study the blow up behavior of the heat equation u(t) = u(xx) with u(x) (0, t) = u(p) (0,t), u(x) (a,t) = u(q) (a,t). We also study the quenching behavior of the nonlinear parabolic equation v(t) = v(xx) +2v(x)(2) /(1-v) with v(x)(0,t) = (1-v(0, t))(-p+2), v(x)(a,t) = (1-v (a, t)(-q+2). In the blow up problem, if u(0) is a lower solution then we get the blow up occurs in a finite time at the boundary x = a and using positive steady state we give criteria for blow up and non-blow up. In the quenching problem, we show that the only quenching point is x = a and v(t) blows up at the quenching time, under certain conditions and using positive steady state we give criteria for quenching and non-quenching. These analysis is based on the equivalence between the blow up and the quenching for these two equations.