当前位置: X-MOL 学术J. Dyn. Diff. Equat. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Approaching Critical Decay in a Strongly Degenerate Parabolic Equation
Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2020-09-02 , DOI: 10.1007/s10884-020-09892-x
Michael Winkler

The Cauchy problem in \({\mathbb {R}}^n\), \(n\ge 1\), for the parabolic equation

$$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$

is considered in the strongly degenerate regime \(p\ge 1\). The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies

$$\begin{aligned} t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \rightarrow \infty \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$(0.1)

The first result of this study complements this by asserting that given any positive \(f\in C^0([0,\infty ))\) fulfilling \(f(t)\rightarrow +\infty \) as \(t\rightarrow \infty \) one can find a positive nondecreasing function \(\phi \in C^0([0,\infty ))\) such that whenever \(u_0\in C^0({\mathbb {R}}^n)\) is radially symmetric with \(0< u_0 < \phi (|\cdot |)\), the corresponding minimal solution u satisfies

$$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$

Secondly, (\(\star \)) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data \(u_0\). It is shown that if the connected components of \(\{u_0>0\}\) comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to (\(\star \)) satisfies

$$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$

Under a somewhat complementary hypothesis, particularly fulfilled if \(\{u_0>0\}\) contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.



中文翻译:

在强退化的抛物方程中逼近临界衰减

\({\ mathbb {R}} ^ n \)\(n \ ge 1 \)中的抛物方程式的柯西问题

$$ \ begin {aligned} u_t = u ^ p \ Delta u \ qquad \ qquad(\ star)\ end {aligned} $$

被认为是在高度退化的状态\(p \ ge 1 \)中。首先关注正连续和有界初始数据的情况,在这种情况下,已知存在最小正经典解,并且该解满足

$$ \ begin {aligned} t ^ \ frac {1} {p} \ Vert u(\ cdot,t)\ Vert _ {L ^ \ infty({\ mathbb {R}} ^ n)} \ rightarrow \ infty \ quad \ hbox {as} t \ rightarrow \ infty。\ end {aligned} $$(0.1)

这项研究的第一个结果通过断言,给定任何正\(f \ in C ^ 0([0,\ infty))\)满足\(f(t)\ rightarrow + \ infty \)\(t \ rightarrow \ infty \)可以找到一个正的递减函数\(\ phi \ in C ^ 0([0,\ infty))\)这样,每当\(u_0 \ in C ^ 0({\ mathbb {R} } ^ n)\)\(0 <u_0 <\ phi(| \ cdot |)\)径向对称,u满足相应的最小解

$$ \ begin {aligned} \ frac {t ^ \ frac {1} {p} \ Vert u(\ cdot,t)\ Vert _ {L ^ \ infty({\ mathbb {R}} ^ n)}} {f(t)} \ rightarrow 0 \ quad \ hbox {as} t \ rightarrow \ infty。\ end {aligned} $$

其次,考虑(\(\ star \))以及涉及非负但不一定严格为正的有界和连续初始数据\(u_0 \)的初始条件。结果表明,如果\(\ {u_0> 0 \} \)的连通分量符合反映​​某些统一有界性的条件,则(\(\ star \))的唯一确定的连续弱解就可以满足

$$ \ begin {aligned} 0 <\ liminf _ {t \ rightarrow \ infty} \ Big \ {t ^ \ frac {1} {p} \ Vert u(\ cdot,t)\ Vert _ {L ^ \ infty ({\ mathbb {R}} ^ n)} \ Big \} \ le \ limsup _ {t \ rightarrow \ infty} \ Big \ {t ^ \ frac {1} {p} \ Vert u(\ cdot,t )\ Vert _ {L ^ \ infty({\ mathbb {R}} ^ n)} \ Big \} <\ infty。\ end {aligned} $$

在某种补充假设下,尤其是如果\(\ {u_0> 0 \} \)包含相关狄利克雷拉普拉斯算子具有任意小的主特征值的分量,则尤其可以实现,最终可以看出(0.1)对于这样的负数也仍然成立弱解。

更新日期:2020-09-03
down
wechat
bug