The solutions of a hyperbolic–parabolic mixed type equation on half-space domain

The solutions of the hyperbolic–parabolic mixed type equation∂u∂t=ΔA(u)+div(b(u)),inQT=R+N×(0,T), are considered, where R+N⊂RN is half-space. A new kind of entropy solution to the equation is introduced. The paper shows that the convection term div(b(u))div(b(u)) determines the explicit boundary value condition. If bN′(0)<0, we can impose the general Dirichlet boundary conditionu(x,t)=0,(x,t)∈∂R+N×(0,T)=Σ×(0,T), which is satisfied in a particular weak sense. But if bN′(0)≥0, then no boundary value condition is necessary, the solution of the equation is free from any limitation of the boundary value condition.