Asymptotic behaviors of the solutions to scalar viscous conservation laws on bounded interval corresponding to rarefaction waves

It is concerned that the asymptotic behaviors of the solutions to the initial-boundary value problem for scaler visous conservation laws u(i) + f(u)(z) = u(xx) on [0, 1], with the boundary conditions u(0, t) = u(-) and u(1, t) = u. as well as the initial data u(x, 0)= u(0)(x) satisfying u(0)(0) = u(-) and u(0)(1) = u(+), where f "(u) > 0 for all u under consideration, u(-) < u(+). By means of an elementary L-2 energy method, both the global existence and the asymptotic behavior are obtained without smallness conditions. Moreover, optimal decay rates are also obtained.