Stability and Instability for Solutions of Burgers’ Equation with a Semilinear Boundary Condition

In this paper, we present several results concerning the long-time behavior of positive solutions of Burgers’ equation $u_t = u_{xx} + \varepsilon uu_x $, $0 0$, $u(x,0)$ given, subject to one of two pairs of boundary conditions: (A) $u(0,t) = 0$, $u_x (1,t) = au^p (1,t)$, $t > 0$, or (B) $u(1,t) = 0$, $u_x (0,t) = - au^p (0,t)$, where $0 < p < \infty $. A complete stability-instability analysis is given. It is shown that some solutions can blow up in finite time. Generalizations replacing $\varepsilon uu_x $ by $(f(u))_x $ and $au^p$ by $g(u)$ are discussed.