On Weak Completeness of the Set of Entropy Solutions to a Degenerate Nonlinear Parabolic Equation

In the case of equation (1.2) condition (1.4) coincides with the known Kruzhkov entropy condition [9]. If the function g(u) strictly increases then any weak solution of (1.1) is an entropy solution of this equation as well (cf. [2]) but for degenerate equations this may be violated and, in particular, entropy condition (1.4) is necessary for the uniqueness of solution to the Cauchy problem for equation (1.1). Remark also that relation (1.3) readily follows from entropy condition (1.4) with k = ±M , where M ≥ ‖u‖∞. Now we consider a bounded in L(ΠT ) sequence un = un(t, x) of entropy solutions of (1.1) weakly convergent to u = u(t, x) ∈ L(ΠT ). In the case of a conservation law (1.2) with the flux φ(u) ∈ C(R) it is rather well known that u = u(t, x) is a weak