ASYMPTOTIC-BEHAVIOR FOR SCALAR VISCOUS CONSERVATION-LAWS WITH BOUNDARY EFFECT

We consider the asymptotic stability of viscous shock wave phi for sca lar viscous conservation laws u(t) + f(u)(x) = u(xx) on the half-space ( -infinity, 0) with boundary values u\(x) = (-x), u\(x = -infinity) = u(+). Our problem is divided into three cases depending on the sign of shock speed s of the shock (u(-), u(+)). When s less than or equal to 0, the asymptotic state of u becomes phi(. + d(t)), where d(t) depe nds implicitly on the initial data u(x,0) and is related to the bounda ry layer of the solution at the boundary x = 0. The stability of this state for s < will be shown by applying the weighted energy method. Fo r s = 0 a conjecture will be shown by applying the weighted energy on ci(t) will be presented. The case s > 0 is also treated. (C) 1997 Acad emic Press