论文标题
Caffarelli-Kohn-Nirenberg重量的快速扩散方程的解决方案的良好特性
Fine properties of solutions to the Cauchy problem for a Fast Diffusion Equation with Caffarelli-Kohn-Nirenberg weights
论文作者
论文摘要
我们研究了与权重的快速扩散方程（$ u_t = | u_t = | x | x |^γ\ mathrm {div} \ left（| x | x |^{ - β{ - β} \ nabla u^m \ right）$（0，+infty pos），在$ d \ ge 3 $中，在所谓的良好快速扩散范围内，$ m_c <m <1 $，在参数$γ，β$的范围内，最适合所谓的Caffarelli-Kohn-Nirenberg不平等现象的有效性。询问这种解决方案的表现是一个自然的问题，就像barenblatt $ \ mathfrak {b} $（基本解决方案）：例如，渐近融合，即$ \ | u（t） - \ mathfrak {b} {b}（t）（t）（t） l}^p（\ mathbb {r}^d）} \ xrightArrow [] {t \ to \ infty} 0 $，以所有$ 1 \ le p \ le \ infty $而闻名，而只有少数部分结果可以解决对尾巴行为的细微分析。我们表征了最大的数据集$ \ MATHCAL {X} \ subset {\ rm L}^1 _+（\ Mathbb {r}^d）$，该$产生的解决方案被刻在两个Barenblatt（Global Harnack Princtiple）之间，并在相对错误（UREC）中汇聚（UREC），i.e.e.e. $ { d} _ \ infty（u（t））= \ | u（t）/\ Mathcal {b}（t）-1 \ | _ {{\ rm l}^\ infty（\ mathbb {r}^d）}这种表征是根据$ u（t = 0）$的整体条件而言。据我们所知，线性加热方程的类似问题$ m = 1 $，没有那么明确的答案。我们的特征对于经典，非加权的FDE也是新的。我们能够以不同的规范提供最小的收敛速率。在非加权情况下，这种速率几乎是最佳的，并且对于径向溶液而言是最佳的。为了完成全景，我们表明具有$ {\ rm l}^1 _+（\ mathbb {r}^d）\ setMinus \ Mathcal \ Mathcal {x} $的解决方案，保留所有时间相同的“脂肪”空间尾巴，因此Urec失败了。
We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^γ\mathrm{div}\left(|x|^{-β}\nabla u^m\right)$ posed on $(0,+\infty)\times\mathbb{R}^d$, with $d\ge 3$, in the so-called good fast diffusion range $m_c<m<1$, within the range of parameters $γ, β$, optimal for the validity of the so-called Caffarelli-Kohn-Nirenberg inequalities. It is a natural question to ask in which sense such solutions behave like the Barenblatt $\mathfrak{B}$ (fundamental solution): for instance, asymptotic convergence, i.e. $\|u(t)-\mathfrak{B}(t)\|_{{\rm L}^p(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$, is well known for all $1\le p\le \infty$, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $\mathcal{X}\subset{\rm L}^1_+(\mathbb{R}^d)$ that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (UREC), i.e. ${\rm d}_\infty(u(t))=\|u(t)/\mathcal{B}(t)-1\|_{{\rm L}^\infty(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$. Such characterization is in terms of an integral condition on $u(t=0)$. To the best of our knowledge, analogous issues for the linear heat equation $m=1$, do not possess such clear answers. Our characterization is also new for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to $\mathcal{B}$ in different norms. Such rates are almost optimal in the non weighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in ${\rm L}^1_+(\mathbb{R}^d)\setminus\mathcal{X}$, preserve the same "fat" spatial tail for all times, hence UREC fails.
