AI 中文总结
该研究在黎曼流形上针对莱本松方程，分q(p-1)≥1、<1两种情况证明解的质量守恒，指出双曲空间上q(p-1)<1时解无质量守恒，还证明了L^(p-1)-刘维尔性质，部分回答了I. Holopainen的猜想。
AI 中文摘要
我们在黎曼流形M上考虑莱本松方程：∂ₜu=Δₚuᵠ，其中p>1且q>0。当q(p-1)≥1时，我们仅假设对某x₀∈M及所有足够大的r>0，体积界V(x₀,r)≤exp(Cr^(p/(p-1)))，就证明莱本松方程解的质量守恒；当q(p-1)<1时，我们假设V(x₀,r)≤Crᴺ且p>N[1-q(p-1)]，该阈值与ℝⁿ中N=n的情况一致。我们还证明双曲空间ℍⁿ上的解在q(p-1)<1时具有有限灭绝时间，这表明质量守恒性质不成立。利用q(p-1)=1时的质量守恒结果，我们还证明了一个L^(p-1)-刘维尔性质，部分回答了I. Holopainen在文献[holopainen2000sharp]中提出的猜想。
英文摘要
We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=Δ_{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.
Comments18 pages
