Behaviors of Solutions for the Burgers Equation with Boundary corresponding to Rarefaction Waves

Abstract

We investigate the asymptotic behaviors of solutions of the initial-boundary value problem to the generalized Burgers equation $u_t + f(u)_x = u_{xx}$ on the half-line with the conditions $u(0,t)=u_-, \quad u(\infty,t)=u_+$, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic state. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds $f'(u_{\pm})$ of the boundary state $u_- = u(0)$ and the far field state $u_+ = u(\infty)$. In all cases both global existence of the solution and the asymptotic behavior are shown without smallness conditions. New wave phenomena are observed. For instance, when $f'(u_-) < 0 < f'(u_+)$, the solution behaves as the superposition of (a part of) a viscous shock wave as boundary layer and a rarefaction wave propagating away from the boundary.