Behavior of solutions of Burgers’s equation with nonlocal boundary conditions. II

We study the large-time behavior of positive solutions of Burgers’s equation ${u_t} = {u_{xx}} + \varepsilon u{u_x}, 0 < x < 1, t > 0\left ( {\varepsilon > 0} \right )$, subject to the nonlocal boundary condition: $- {u_x}\left ( {0, t} \right ) - \frac {1}{2}\varepsilon {u^2}\left ( {0, t} \right ) = a{u^p}\left ( {0, t} \right ){\left ( {\int _0^1 u \left ( {x, t} \right )dx} \right )^q},u\left ( {1, t} \right ) = 0 \\ \left ( {0 < p, q < \infty } \right )$. The steady-state problem is analyzed in detail, and the result about finite-time blow-up is proved.