The Mixed Boundary Value Problem for Degenerate Quasilinear Parabolic Equations

<正> Let Ω/R~n（n≥2）be a bounded open set;QT=Ω×[0,T],ST=Ω×[0,T],S1,S2 bethe partial boundaries of Ω and S1∪S2=Ω,S1∩S2=.We denote Γ1,T=S1×[0,T],Γ2,TS2×[0,T],and consider the problem(u_t-/（xi）ai（x,t,u,ux）+a（x,t,u,ux）=f（x,t）,（x,t）∈QT,αi（x,t,u,ux）cos（,xi）=（x,t）,（x,t）∈Γ1,Tu（x,t）=0 （x,t）∈Γ2,Tu（x,0）=0（x）,x∈Ωwhere ai（x,t,u,ux）is not necessarily monotonic as usually is.Some other conditions are also weakened.Under such mild conditions we obtain the existence of a generalized solution of the mixed boundary valueproblem for the above degenerate parabolic equation.We still employ the Galerkin method and some techiques to prove the main lemma.The method usedto prove our theorem is still the one initiated by us before.Finally,an example is given to illustrate the application of our theorem in the non-steady permeatingproblem originating in hydrokinetics.