Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

Abstract   This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u _t + f(u) _x = u _x,x on 0,1], with the boundary condition u(0,t)=u ₋,u(1t)=u ₊ and the initial data u(x,0)= u ₀(x), where u ₋ ≠ u ₊ and f is a given function satisfying f"u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When u ₋ < u ₊, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u ₋ > u ₊, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |u ₋ − u ₊| is small. Moreover, exponential decay rates are both given. *Partially supported by the National Natural Sciences Foundation of China (No. 10101014), the Key Project of Natural Sciences Foundation of Beijing and Beijing Education Committee Foundation. **Supported by the National Natural Science Foundation of China (No. 10061001) and Guangxi Natural Science Foundation (No. 9912020).