| Abstract: | Consider the initial boundary value problem of the strongdegenerate parabolic equation$$partial_{xx}u+upartial_{y}u-partial_{t}u =f(x,y,t,u),quad(x,y,t)in Q_{T}=Omegatimes (0,T)$$with a homogeneous boundary condition. By introducing a new kind ofentropy solution, according to Oleinik rules, the partial boundarycondition is given to assure the well-posedness of the problem. Bythe parabolic regularization method, the uniform estimate of thegradient is obtained, and by using Kolmogoroff''s theorem, thesolvability of the equation is obtained in $BV(Q_{T})$ sense. Thestability of the solutions is obtained by Kruzkov''s double variablesmethod. |
