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ABSTRACT. Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spaces Loo(fl) or L2(fl) and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved i i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.KEY WORDS: strong stability of solutions, generalized solution, equation of motion of a gas, viscous heatconducting polytropic gas, quasilinear system of composite type, initial boundary value problem.Systems of equations governing the motion of a viscous gas form an interesting and important class of quasilinear systems of equations of composite type [1,2]. Results on the "global" existence and uniqueness of the regular generalized solutions of initial boundary value problems for equations of motion of a viscous heat-conducting gas with initial data from Wl(fl) is outlined in [2, Chap. 2]. A theorem on stability of such solutions in the strong norm is given in [3]. The stability of regular generalized solutions was studied in detail in [4,5] (in the finite difference version, but the same techniques apply in the differential case).The "global" existence of (weak) generalized solutions for these problems with initial data from Lq(~) with some q was obtained in [6,7] Under somewhat more rigid conditions on the initial data, uniqueness and stability of generalized solutions was also established in [6]. An earlier result on stability is contained in [8 I. It should be noted that the weaker the class of generalized solutions, the more complicated is the proof c r uniqueness and stability in it.Earlier, in [9], the authors proved the strong stability of generalized solutions for systems of equations of motion of a viscous barotropic gas. (Strong stability implies the possibility of expressing an estimate for the norm of the difference of two solutions in terms of the sums of the norms of the differences of the corresponding data.) The present paper can be regarded as a natural continuation and extension of these studies. In it we establish the strong stability of generalized solutions from [6] for systems of equations of motion of a viscous heat-conducting gas without imposing any additional conditions on the data. Three initial boundary value problems with inhomogeneous boundary conditions are studied. Uniqueness of generalized solutions from [6] is shown to be a simple consequence of stability. Moreover, the proof of one of the two results on (strong) stability announced in [6] is given. The proof essentially uses the techniques and auxiliary results outlined in [9], as well as one result from [7]. Only the new features of the proofs bearing on the specific character of t...

We consider the following nonlinear initial-boundary value problem, which, in particular, describes a nonstationary radiative heat transfer in a system of moving absolutely black bodies:where the G j are bounded strictly Lipschitz domains in R N (N ≥ 2) such thatḠ i ∩Ḡ j = ∅ for i = j. We use the notation D t = ∂/∂t; moreover, n is the unit outward normal on ∂G, and dσ is a natural measure on ∂G.In the radiative heat transfer problem, x is a Lagrange Cartesian coordinate (x ∈ R N , N = 2, 3), t is time, the unknown function u(x, t) has the physical meaning of absolute temperature, f (x, t) and g(x, t) are the heat source and heat flux densities, h(u) is the surface radiation flux density [h(u) = κ|u| 3 u corresponds to the Stefan-Boltzmann radiation law], and ∂G h(u(ξ, t))ϕ(ξ, x, t)dσ(ξ) is the radiation flux density absorbed at a point x.The function ϕ (the angular coefficient) is defined on ∂G × ∂G × (0, T ). If the system of bodies is at rest, then the angular coefficient ϕ is independent of t and has the formIn the general case of moving bodies, the Euler coordinate x e of a point x ∈ ∂G and the outward normal n e to ∂G at x are functions of x and t, and ϕ acquires the form(1.4) here r = x e (x, t) − x e (ξ, t), and G(t) is the domain filled by the system of bodies at time t. Problem (1.1)-(1.3) is characterized by the fact that the boundary condition (1.2) is nonlocal and nonlinear. Moreover, the function h(u) can have a nonlinearity with respect to u of order higher than that admitted by embedding theorems.