Abstract: We study the limit of the hyperbolic-parabolic approximation

v^ε_t+A(v^ε,εv^ε_x)v^ε_x = εB(v^ε)v^ε_{xx},

with data β(v^ε(t,0)) = g and v^ε(0,x) = v₀. The function β is defined in such a way to guarantee that the initial boundary value problem is well posed even if B is not invertible. The data g and v₀ are constant.

When B is invertible, the boundary datum can be assigned by imposing v^ε(t,0) = v_b, where v_b is again a constant. The conservative case is included in the previous formulations.

It is assumed convergence of the v^ε, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit.

The most interesting points are the following two.

First, the boundary characteristic case is considered, i.e. one eigenvalue of A can be 0.

Second, as pointed out before we take into account the possibility that B is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.