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 \）为\（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）\）与\（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。
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