Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws Original Research Article

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = uxx on [0, 1], with the boundary condition u(0, t) = u−(t) → u−, u(1, t) = u+(t) → u+, as t → +∞ and the initial data u(x,0) = u0(x) satisfying u0(0) = u−(0), u0(1) = u+(1), where u± are given constants, u− ≠ u+ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, 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, that is, |u−−u+| is small. Moreover, when u±(t) ≡ u±, t ≥ 0, exponential decay rates are both obtained.