The solutions of the hyperbolic-parabolic mixed type equationare considered, where R N + ⊂ R N is half-space. A new kind of entropy solution to the equation is introduced. The paper shows that the convection term div(b(u)) determines the explicit boundary value condition. If b N (0) < 0, we can impose the general Dirichlet boundary conditionwhich is satisfied in a particular weak sense. But if b N (0) ≥ 0, then no boundary value condition is necessary, the solution of the equation is free from any limitation of the boundary value condition.

Consider a parabolic equation related to the p-Laplacian. If the diffusion coefficient of the equation is degenerate on the boundary, no matter we can define the trace of the solution on the boundary or not, by choosing a suitable test function, the stability of the solutions always can be established without a boundary condition. MSC: 35K55; 46E35; 35R35

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Consider the equation