LAUSR

A degenerate parabolic equation of the form ( | v | β − 1 v ) t = div ( b ( x , t ) | ∇ v | p ( x , t ) − 2 ∇ v ) + ∇ g → ⋅ ∇ γ → ( v ) $$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$ is considered, where g → = { g i ( x , t ) } $\vec{g}=\{g^{i}(x,t)\}$ , γ → ( v ) = { γ i ( v ) } $\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$ . If the diffusion coefficient b ( x , t ) ≥ 0 $b(x,t)\geq0$ is degenerate on the boundary, by adding some restrictions on b ( x , t ) $b(x,t)$ and g⃗ , the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.