根据YIN和WANG的方法,结合Fichera-Oleinik理论,研究奇异扩散方程：∂φ( u)/∂t =div(ραÑu p-2Ñu),(x,t)∈QT =Ω×(0,T),其中Ω是RN 中的有界区域,边界∂Ω充分光滑,ρ(x)=dist(x,∂Ω), p >1,α>0,φ满足：φ∈C2,且存在δ>0使得φ′(s)>δ>0.证明了α≥p -1时,不需要任何边值条件,方程最多有一个满足初值条件的解；而0<α< p -1时,方程存在唯一满足初边值条件弱解.%The paper studies the singular diffusion equation in the method of YIN 's and WANG 's with Fichera-Oleinik theory:∂φ(u)/∂t = div(ρα Ñu p-2 Ñu),(x,t)∈QT = Ω× (0,T), whereΩis a bounded domain in RN with appropriately smooth boundary∂Ω,ρ(x) = dist(x,∂Ω) , p > 1,α > 0 ,φ∈C2 , and there existsδ >0 such thatφ′( s) > δ >0 . The paper proves that ifα≥p-1 , the equation admits a unique solution subject only to a given initial condition without any boundary condition, while if 0 < α < p -1 , for a given initial condition, the equation admits different solutions for different boundary conditions.
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