The paper reviews Fichera-Oleinik theory, then uses the theory to discuss the boundary value condition related to the equation.
Instead of (1.11), the condition (1.13) has the degeneracy on the boundary independent of the boundary value condition.
It follows from the boundary value condition (1.3) that the boundary value condition (1.2) can be replaced by ∇ Δ u ( x, t ) ⋅ ν = 0, x ∈ ∂ Ω, t ∈ ( 0, T ).
Then the problem (2.1) with the boundary value condition (2.2) has a unique positive solution u ∗ ∈ P h.
In other words, the solution of the equation is completely controlled by the initial value condition.
The test function chosen to verify the uniqueness of the solutions should be independent of the boundary value condition.
end{aligned} (1.6) Besides the initial value condition (1.2), instead of the usual boundary value condition (1.3), by assumptions (1.5 - 1.6), only a partial boundary value condition u x,t)=0, quad x,t inSigma_{1}times 0,T) (1.7) should be imposed.
on, then is the Green's function of the differential equation in with respect to the boundary value condition.
Let (u x,t)) be the entropy solution of equation (1.1) with the initial value condition (1.3).
In other words, the solution of equation (1.1) is completely controlled by the initial value condition.
If (Sigma_{1}=emptyset), the solution of equation (1.1) is completely controlled by the initial value condition.
