Asymptotic stability of solution to the initial-boundary value problem for scalar viscous conservation laws corresponding to rarefaction waves (Q2721897)

The authors investigate the scalar viscous conservation law \(u_t+f(u)_x=u_{xx}\), \(x>0\), \(t>0\) with boundary and initial conditions \(u(0,t)\to u_-\) as \(t\to+\infty\) resp. \(u(x,0)\to u_+\) as \(x\to+\infty\). It is supposed that \(f''(u)>0\) and \(u_-<u_+\). The problem is divided into five cases depending on the signs of the characteristic speeds \(f'(u_\pm)\). Both the global existence of the solution and the asymptotic stability are shown in all cases. The asymptotic states are described by means of rarefaction waves from the corresponding Riemann problem \(u_t+f(u)_x=0\), \(u(0,x)=u_\pm\) and viscous shock waves from stationary solution to the equation \(f(v)_x=v_{xx}\) connecting \(u_-\) to \(u_+\) or 0.