By Curtiss D.R.

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Such a procedure may be rigorously justified both for the Cauchy problem (cf. 2]). 6), boundedness of on in the space L2(Q). 17) for any b E C' [0, oo) globally Lipschitz on [0, oo). 4. 12), no matter how strong it seems, is satisfied in a number of important cases. 2) with f (t, x) = g(t, x) + VF(x) where sup max Ige(t, x) I E L1(0, oo). 3) that IVuI2 dx dt is finite. Consequently, the sequence of the time-shifts on (t, x) = o(t + n, x), un(t, x) = u(t + n, x) 32 E. 1 for any finite interval I, say, I = (0, 1).

Vii) Moreover, if such a function exists, it is given by the formula o(x) = exp(F(x K a- if 'y = 1 or LO (x) _ (yay l) ([F(x) - K)+) for y > 1 where K is a certain constant depending on M. 2. Let y = 1, F locally Lipschitz and bounded from below on St and on meas(1t) = oo. 2) with a finite and strictly positive mass M. 28 4 E. Feireisl Convergence of global trajectories In this section, we state a result on convergence of global solutions, provided they exist, of course. Unfortunately, we are able to treat only the case when the equation of state takes the form p(o)=ap , a>0, 7> 3 2 We remark, however, that the case -y E [1, 2] could be handled in exactly the same way, provided suitable a priori estimates of the density o were available.

1). If f is not continuous but LP-Caratheodory, this last result does not hold anymore. e. 1) to be tangent to the curve u = a(t) from above. This is, for example, the case for the bounded function f(t,u,v) -1 u2 + cost 1 + cost if u < -1, if - 1 < u < cost, t # ir, _ - cost if cost < u, if we consider a(t) -1, u(t) = cost, a = 0 and b = 21r. This remark motivates the following proposition. 3. Let f : [a, b] x R2 - JR be an L' -Caratheodory function. e. t E Io, for all u, v, with a(t) < u < a(t) + co, al(t) - co < v < a'(t) + Eo, we have a"(t) > f (t, u, v); (b) a(a) > 0, a'(b) < 0.