We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t}-\operatorname{div}\left(m|u|^{m-1} \nabla u\right)=f \in L^{q, r}, \quad m>1$$ are locally Hölder continuous, with exponent $$\gamma = \min \{ {{\alpha _0^ - } \over m},\;{{[(2q - n)r - 2q]} \over {q[(mr - (m - 1)]}}\} ,$$ where α 0 denotes the optimal Hölder exponent for solutions of the homogeneous case. The proof relies on an approximation lemma and geometric iteration in the appropriate intrinsic scaling.

中文翻译：

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非均匀多孔介质方程的锐性规律

我们证明了非均匀多孔介质方程 $$u_{t}-\operatorname{div}\left(m|u|^{m-1} \nabla u\right)=f \in L^{ 的局部有界解q, r}, \quad m>1$$ 是局部 Hölder 连续的，指数 $$\gamma = \min \{ {{\alpha _0^ - } \over m},\;{{[(2q - n )r - 2q]} \over {q[(mr - (m - 1)]}}}\} ,$$ 其中 α 0 表示齐次情况的解的最优 Hölder 指数。证明依赖于近似引理和适当的内在缩放中的几何迭代。