In this paper, we present several results concerning the long time behavior of positive solutions of Burgers' equation u[subscript] t = u[subscript]xx+[epsilon] uu[subscript]x,[epsilon]>0,00,u(x,0) given, subject to one of four pairs of boundary conditions:(UNFORMATTED TABLE OR EQUATION FOLLOWS) (A[subscript]1) u(0,t) = 0,u[subscript] x(1,t) = a(1 - u(1,t))[superscript]-p, t > 0, &(B[subscript]1) u(1,t) = 0,u[subscript]x(0,t) = -a(1 - u(0,t))[superscript]-p, t > 0, &(C[subscript]1) u(0,t) = 0,u[subscript]x(1,t) = a[over] u[superscript]p(1,t), t > 0, & or &(D[subscript]1) u[subscript]x(0,t) = -a[over] u[superscript]p(1,t), u(1,t) = 0, t > 0, & where 0 0. (TABLE/EQUATION ENDS);A complete stability-instability analysis is given. It is shown that for (A) and (B) some solutions quench (reach one in finite time) and that when this happens u[subscript] t(1,t) blows up at the same time. Generalizations replacing uu[subscript] x by (f(u))[subscript]x and (1 - u)[superscript]-p or a[over] u[superscript] p(1,t) by g(u) are discussed with special emphasis on the case g(u) = au[superscript] p - [epsilon][over] 2 u[superscript]2.