Exponential Stability Behavior For A One-Dimensional Isentropic And Isothermal Model System Of Compressible Viscous Gas

In this paper, we discuss the global existence and exponential stability behavior forone-dimensional isentropic and isothermal model system of compressible viscous gas in abounded region with an external force and initial conditions as well as boundary conditions.The model we consider is as follows: vt ? ux = 0, ut + (av?1)x =μ(uvx )x + f( 0 xv dy,t)with initial conditions v(x,0) = v0(x), u(x,0) = u0(x) and boundary conditions u(0,t) =u(1,t) = 0.In the same model, Mucha[7] obtained the global existence and exponential stabilityof the solutions to the problem under other boundary conditions and when the state func-tion is:p(v) = av-γ, (a > 0,is constant) andγ> 1. Yanagi[19] obtained the existencebehavior of periodic solutions to the problem when the state function is:p(v) = av-γ, (a >0,is constant) andγ≥1. In [22], when the external force is f(ξ,τ) = f∞(ξ) + f(ξ,τ),Zhang and Fang obtained the global existence behavior of solutions to the problem. In[5],A. Matsumura and T. Nishid proved the existence of the priori solutions whenγ= 1. Theproof of this paper is based on the methods used by Y. Qin in references [8-14], in whichhe proved the global existence and exponential stability of solutions for one dimensionalmodel.