A degenerate parabolic equation of the form $$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$ is considered, where $\vec{g}=\{g^{i}(x,t)\}$, $\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$. If the diffusion coefficient $b(x,t)\geq0$ is degenerate on the boundary, by adding some restrictions on $b(x,t)$ 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.

中文翻译：

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变指数增长阶简并抛物方程的非负弱解

$$ \ bigl（\ vert v \ vert ^ {\ beta-1} v \ bigr）_ {t} = \ operatorname {div} \ bigl（b（x，t）\ vert \ nabla v \ vert ^ {p（x，t）-2} \ nabla v \ bigr）+ \ nabla \ vec {g} \ cdot \ nabla \ vec {\ gamma}（v）$$，其中$ \ vec {g} = \ {g ^ {i}（x，t）\} $，$ \ vec {\ gamma}（v）= \ {\ gamma_ {i}（v）\} $。如果扩散系数$ b（x，t）\ geq0 $在边界上退化，则通过对$ b（x，t）$和g⃗施加一些限制，证明了弱解的存在性和唯一性。基于唯一性，可以在没有任何边界条件的情况下证明弱解的稳定性。