We study a nonlinear elliptic degenerate equation with the form: b(u)−a(u,φ(u)x)x=f which is associated to the elliptic-parabolic-hyperbolic equation of the form: b(u)t−a(u,φ(u)x)x=f. We prove in this work without Alt and Luckhaus structure condition existence and uniqueness of entropy solutions of the associated Dirichlet problem. We also define an operator associated to the evolution problem and prove some useful properties of this operator.

elliptic; parabolic; hyperbolic; weak solution; entropy solution