AI 中文总结
研究无限加权图上广义多孔介质方程柯西问题，通过建立原理和论证构造解，对特定非线性推导能量估计，得出不同情况下的结果，还证明了随机完备图上非负\(\ell^1 -\)温和解的广义质量平衡。
AI 中文摘要
我们研究无限加权图上广义多孔介质方程的柯西问题。对于一般非线性，在有限子图上建立狄利克雷比较和弱最大值原理，并通过穷竭论证，为任意\(\ell^\infty\)初始数据构造最小和最大逐点解，由显式的、可能依赖时间的障碍控制。对于多孔非线性\(\phi(s)=s|s|^{m - 1}\)，假设\(\nu>2\)的\(\nu -\)索伯列夫不等式，我们推导\(\ell^1 -\)温和解的定量能量估计。这在快速扩散范围\(0<m<2/\nu\)产生有限时间熄灭，在范围\(m>2/\nu\)产生\(\ell^1 -\)到\(\ell^q\)平滑。最后，我们证明了在无穷远处随机完备的图上非负\(\ell^1 -\)温和解的精确广义质量平衡，允许任意的杀伤项。对于合适的经典和有界逐点解也建立了相同的平衡。在没有杀伤的情况下，这些归结为质量守恒。
英文摘要
We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $ϕ(s)=s|s|^{m-1}$, assuming a $ν$-Sobolev inequality with $ν>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0<m<2/ν$ and $\ell^1$-$\ell^q$ smoothing in the range $m>2/ν$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.
