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

✍ Scribed by Tao Pan; Hongxoa Liu

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 490 KB
• Volume: 15
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(02)00034-4

✦ Synopsis

This paper is concerned with the asymptotic behaviors of the solutions to the initialboundary value problem for scalar viscous conservations laws ut + f(u), = uzz on [0, 11, with the boundary condition u(O,t) = u_(t) -+ u_, u(l,t) = u+(t) + u+, as t --t +m and the initial data u(z,O) = uo(z) satisfying uo(O) = u-(O), un(l) = u+(l), 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, Izl_ -u+I is small. Moreover, when u*(t) = u*, t 2 0, exponential decay rates are both obtained.