AI 中文总结
研究一般度量测度空间上带符号多孔介质方程柯西问题的适定性，基于Rothe方法和单调算子理论，仅用狄氏型定义，证明对给定初始数据存在唯一弱解，适用于多种度量测度空间包括非光滑分形。
AI 中文摘要
我们在一般度量测度空间上，为带符号的多孔介质方程及其快速扩散对应方程的柯西问题发展了一个适定性理论。方程为\(\partial_t u=\mathcal{L}\left(|u|^{m - 1}u\right)\)，\(m>0\)，其中\(\mathcal{L}\)是对称狄氏型的生成元。证明了对于每个初始数据\(u_0\in L^{m + 1}(M,\mu)\)，存在唯一函数\(u\)在合适意义下弱解该方程。证明基于Rothe方法和单调算子理论，仅使用狄氏型定义及其扩展版本，无需空间形式或几何的额外性质，该理论适用于广泛的度量测度空间，包括非光滑分形。
英文摘要
On general metric measure spaces, we develop a new well-posedness theory for the signed porous medium equation and its fast diffusion counterpart \[ \partial_t u = \mathcal{L}\left(|u|^{m-1}u\right), \qquad m>0, \] where $\mathcal{L}$ is the associated non-positive self-adjoint operator of a symmetric Dirichlet form. The theory does not rely on a Gelfand triple or compact embeddings; instead, it is built upon the extended Dirichlet space $\mathcal{F}_e$ and auxiliary spaces $V^q:=L^q\cap\mathcal{F}_e$, whose uniform convexity plays a key role in the proof. The proof uses only the existence of the Dirichlet form and its extension; no additional regularity of the form or geometric assumptions on the underlying space are needed. Consequently, the results apply to a wide range of metric measure spaces, including non-smooth fractals.
Comments39 pages, 2 figures. Have corrected nearly all errors in old versions. Comments are welcome
