Compressible Navier-Stokes Equations With A Non-autonomous External Force And A Heat Source

In this paper, we prove the global existence and asymptotic behavior, as time tends to infinity, of solutions in H² to the initial boundary value problem of the compressible Navier-Stokes equations of one-dimensional motion of a viscous heat conducting gas in a bounded region with a non-autonomous external force and a heat source. Some new ideas and more delicate estimates are used to prove these results. First, we establish the expression of u and then obtain its uniform positive lower and upper bound. Then, we bound the norm of specific volume, velocity, temperature as well as their derivatives, in terms of the expression 1 + sup_0≤s≤t ||θ（t）||_l^∞ and then get the global existence in H¹; We obtain the the uniform global （in time） positive lower boundedness （independent of t） of the absolute temperatureθ, which combines with a series of inequalities can be get the global existence in H². Finally, we obtain the asymptotic behavior by using an important lemma.