\begin{thm}\label{thm:lower}
Fix an integer $k\ge 3$. Then, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{c_k-o(1)},
\qquad
c_k:=\frac{k-2}{e ((k-2)!)^{1/(k-2)}}.
\]
\end{thm}

\begin{lem}\label{lem:harmonic-SS}
Let
\[
H_\ell(N) := \sum_{\substack{n\le N\\ \omega(n)=\Omega(n)=\ell}} \frac1n.
\]Then for every $0<\eta<1$ we have
\begin{equation}\label{eq:H-ell-lower}
  H_\ell(N) \gg_\eta \frac{(\log\log N)^\ell}{\ell!}
  \qquad (N\to\infty,\ 1\le \ell\le (1-\eta)\log\log N).
\end{equation}
\end{lem}

\subsection{The problem and background}Fix integers $k\ge 3$ and $N\ge 1$. Following Erd\H{o}s~\cite{Er70}, we study subsets
$A\subseteq [N]:=\{1,2,\dots,N\}$ that avoid certain ``equal--LCM'' configurations.
We say that $A$ \emph{contains an LCM-$k$-tuple} if there exist distinct
$a_1,\dots,a_k\in A$ such that all pairwise least common multiples coincide, i.e.
\[
\operatorname{lcm}(a_i,a_j)=\operatorname{lcm}(a_1,a_2)\qquad (1\le i<j\le k).
\]
We say $A$ is \emph{LCM-$k$-free} if it contains no LCM-$k$-tuple.

In~\cite[p.~124]{Er70}, Erd\H{o}s introduced the counting function $g(k,x)$, defined as
the largest integer $s$ for which one can find
\[
1\le a_1<\cdots<a_s\le x
\]
with the property that no $k$ of the $a_i$ form an LCM-$k$-tuple. Erd\H{o}s
originally conjectured that $g(k,x)=o(x)$ for every $k\ge 3$, but later proved that
this fails for all $k\ge 4$; see~\cite[Theorem~1]{Er70}. In particular, for each
$k\ge 4$ there exists $\delta_k>0$ such that
\[
g(k,x)\ge \delta_k x
\]
for all sufficiently large $x$.

Erd\H{o}s also considered a harmonic analogue. He showed~\cite[Eq.~(19)]{Er70} that for each
$k\ge 3$ there exists $b_k>0$ such that whenever $x$ is sufficiently large and
$1\le a_1<\cdots<a_s\le x$ satisfy
\begin{equation}\label{eq:Erdos-threshold}
\sum_{i=1}^s \frac{1}{a_i} > b_k \log x,
\end{equation}
then some $k$ of the $a_i$ form an LCM-$k$-tuple. In~\cite[p.~127]{Er70}, Erd\H{o}s wrote:
\begin{quote}
\emph{I do not know how much \eqref{eq:Erdos-threshold} can be weakened so that there should always
be $k$ $a$'s every two of which have the same least common multiple.}
\end{quote}
If we define
\[
f_k(N)\;:=\;\max\left\{\sum_{a\in A}\frac1a:\ A\subseteq [N]\ \text{is LCM-$k$-free}\right\}.
\]
then Erd\H{o}s is asking the following question.
\begin{prob}\label{prob:856}
Let $k\ge 3$. Estimate $f_k(N)$.
\end{prob}

Problem~\ref{prob:856} appears as Problem~\#856 on Bloom's Erd\H{o}s Problems website~\cite{EP856}
and may be viewed as the harmonic counterpart of the counting problem introduced by Erd\H{o}s
in~\cite[p.~124]{Er70}.

A trivial bound is
\[
f_k(N)\le \sum_{n\le N}\frac1n = \log N + O(1).
\]

Erd\H{o}s~\cite{Er70} provided a short proof of the improved upper bound $f_k(N)\ll_k \log N/\log\log N$. It is observed in the comment section on the Erd\H{o}s Problems website~\cite{EP856}
that one can further improve this to $f_k(N)\leq \log N \cdot \exp\left(-\Omega_k\left(\frac{\log\log N}{\log\log\log N}\right)\right)$. To our knowledge, no lower bounds of comparable strength were previously available.

\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.

\subsection{The problem and background}Fix integers $k\ge 3$ and $N\ge 1$. Following Erd\H{o}s~\cite{Er70}, we study subsets
$A\subseteq [N]:=\{1,2,\dots,N\}$ that avoid certain ``equal--LCM'' configurations.
We say that $A$ \emph{contains an LCM-$k$-tuple} if there exist distinct
$a_1,\dots,a_k\in A$ such that all pairwise least common multiples coincide, i.e.
\[
\operatorname{lcm}(a_i,a_j)=\operatorname{lcm}(a_1,a_2)\qquad (1\le i<j\le k).
\]
We say $A$ is \emph{LCM-$k$-free} if it contains no LCM-$k$-tuple.

In~\cite[p.~124]{Er70}, Erd\H{o}s introduced the counting function $g(k,x)$, defined as
the largest integer $s$ for which one can find
\[
1\le a_1<\cdots<a_s\le x
\]
with the property that no $k$ of the $a_i$ form an LCM-$k$-tuple. Erd\H{o}s
originally conjectured that $g(k,x)=o(x)$ for every $k\ge 3$, but later proved that
this fails for all $k\ge 4$; see~\cite[Theorem~1]{Er70}. In particular, for each
$k\ge 4$ there exists $\delta_k>0$ such that
\[
g(k,x)\ge \delta_k x
\]
for all sufficiently large $x$.

Erd\H{o}s also considered a harmonic analogue. He showed~\cite[Eq.~(19)]{Er70} that for each
$k\ge 3$ there exists $b_k>0$ such that whenever $x$ is sufficiently large and
$1\le a_1<\cdots<a_s\le x$ satisfy
\begin{equation}\label{eq:Erdos-threshold}
\sum_{i=1}^s \frac{1}{a_i} > b_k \log x,
\end{equation}
then some $k$ of the $a_i$ form an LCM-$k$-tuple. In~\cite[p.~127]{Er70}, Erd\H{o}s wrote:
\begin{quote}
\emph{I do not know how much \eqref{eq:Erdos-threshold} can be weakened so that there should always
be $k$ $a$'s every two of which have the same least common multiple.}
\end{quote}
If we define
\[
f_k(N)\;:=\;\max\left\{\sum_{a\in A}\frac1a:\ A\subseteq [N]\ \text{is LCM-$k$-free}\right\}.
\]
then Erd\H{o}s is asking the following question.
\begin{prob}\label{prob:856}
Let $k\ge 3$. Estimate $f_k(N)$.
\end{prob}

A trivial bound is
\[
f_k(N)\le \sum_{n\le N}\frac1n = \log N + O(1).
\]

Erd\H{o}s~\cite{Er70} provided a short proof of the improved upper bound $f_k(N)\ll_k \log N/\log\log N$. It is observed in the comment section on the Erd\H{o}s Problems website~\cite{EP856}
that one can further improve this to $f_k(N)\leq \log N \cdot \exp\left(-\Omega_k\left(\frac{\log\log N}{\log\log\log N}\right)\right)$. To our knowledge, no lower bounds of comparable strength were previously available.

\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.

\subsection{A polylogarithmic lower bound}Our first result provides an explicit polylogarithmic lower bound.

Problem~\ref{prob:857} is widely known as the Erd\H{o}s--Szemer\'edi sunflower conjecture~\cite{ErSz78},
and it appears as Problem~\#857 on Bloom's Erd\H{o}s Problems website~\cite{EP857}. Following the
notation of~\cite{NaSa17}, let $F_k(n)$ be the maximum size of a $k$-sunflower-free family
$\mathcal{F}\subseteq 2^{[n]}$, and define the \emph{Erd\H{o}s--Szemer\'edi $k$-sunflower-free capacity}
\[
\mu_k^{\mathrm S}:=\limsup_{n\to\infty} F_k(n)^{1/n}.
\]Trivially $\mu_k^{\mathrm S}\le 2$, and the celebrated Erd\H{o}s--Szemer\'edi sunflower conjecture~\cite{ErSz78} asserts that
$\mu_k^{\mathrm S}<2$ for every $k\ge 3$. A standard tensor power argument\footnote{Here is a concise description of the argument. Let $\mathcal{F}_n\subseteq 2^{[n]}$ be a $k$-sunflower-free family. Choose $r\in\{0,1,\dots,n\}$ maximizing $|\mathcal{F}_n\cap\binom{[n]}{r}|$, and set $\mathcal{F}_n':=\mathcal{F}_n\cap\binom{[n]}{r}$, so that $|\mathcal{F}_n'|\ge \frac{1}{n+1}|\mathcal{F}_n|$. For each integer $t\ge 2$, partition $[tn]$ into $t$ subsets $S_1\sqcup\cdots\sqcup S_t$ with $|S_i|=n$. For each $i$ fix a bijection $\pi_i:[n]\to S_i$ and define a copy of $\mathcal{F}_n'$ on $S_i$ by $\mathcal{F}_n'^{(i)}:=\{\pi_i(F):F\in\mathcal{F}_n'\}\subseteq 2^{S_i}$. Then one checks that $\mathcal{F}_{n}^{\otimes t}:=\{F_1\sqcup\cdots\sqcup F_t:\ F_i\in\mathcal{F}_n'^{(i)}\}\subseteq 2^{[tn]}$ is $k$-sunflower-free and satisfies $|\mathcal{F}_{n}^{\otimes t}|=|\mathcal{F}_n'|^t\ge (|\mathcal{F}_n|/(n+1))^t$.} shows that the limsup can be replaced with a limit:
\begin{equation}
 \label{eq:lim}
 \mu_k^{\mathrm S} =\lim_{n\to\infty} F_k(n)^{1/n}.
\end{equation}

\begin{thm}\label{thm:equivalence}
For each $k\ge 3$, we have $\mu_k^{\mathrm S} = 2$ if and only if $f_k(N) = (\log N)^{1 - o(1)}$.
\end{thm}
The implication ``$\Rightarrow$'' is obtained by constructing a LCM-$k$-free set of squarefree integers
whose prime-support sets form a large cosunflower-free family in an appropriate weighted
product measure. The implication ``$\Leftarrow$'' follows from Theorem~\ref{thm:B-implies-A} below.
\begin{thm}\label{thm:B-implies-A}
Fix $k\ge 3$. Then, as $N\to\infty$,
\[
f_k(N)\ll (\log N)^{\mu_k^{\mathrm S}-1+o(1)}.
\]
\end{thm}
Its proof is a double-counting argument similar to Erd\H{o}s's original argument in~\cite{Er70}. To each integer $m$ we
associate a set system on the prime divisors of $m$ and show that LCM-$k$-freeness forces these set
systems to be $k$-sunflower-free. We then combine the resulting estimate with a Sathe--Selberg type
lower bound for the harmonic sum of squarefree almost primes.

Finally, we wonder how good a lower bound for $f_k(N)$ we can establish given knowledge of $\mu_k^{\mathrm S}$. While we can obtain a lower bound by following the proof of Theorem~\ref{thm:equivalence}, it is not good when $\mu_k^{\mathrm S}$ is smaller than $2$. Towards an improvement on Theorem~\ref{thm:lower}, we show that
\begin{thm}\label{thm:mu-lb-implies-f-lb}
Fix $k\ge 3$. Then, as $N\to\infty$,
\[f_k(N) \ge (\log N)^{\log\mu_k^{\mathrm S} - o(1)}.\]
\end{thm}
For $k=3$, the best current upper bound due to Naslund and Sawin~\cite[Theorem~1]{NaSa17} states that
\[
\mu_3^{\mathrm S}\le \frac{3}{2^{2/3}}=1.889881574\dots\, .
\]
In the opposite direction, a construction of Deuber, Erd\H{o}s, Gunderson, Kostochka and
Meyer~\cite{DEGKM97} yields \(\mu_3^{\mathrm S}>1.551\). Hence, by Theorems~\ref{thm:B-implies-A} and~\ref{thm:mu-lb-implies-f-lb}, we obtain the following consequence.
\begin{cor}\label{cor:f3-upper}
As $N\to\infty$,
\[
(\log N)^{\log(1.551) - o(1)} \leq f_3(N)\ll (\log N)^{\frac{3}{2^{2/3}}-1+o(1)}.
\]
\end{cor}
Unfortunately, Theorem~\ref{thm:mu-lb-implies-f-lb} never gives an exponent greater than $\log 2 < 0.7$, so it is strictly worse than Theorem~\ref{thm:lower} for $k \geq 7$. We leave it as an open question to obtain the correct exponent.
\begin{prob}
 In terms of $\mu_k^{\mathrm S}$, determine the smallest $c > 0$ such that for all sufficiently large $N$, we have
 $$f_k(N) \leq (\log N)^{c + o(1)}.$$
\end{prob}

\medskip\noindent
\textbf{2. Prime bucketing into $t$ disjoint blocks of harmonic sum $\ge 1$.}
Now fix $N$ large (how large will be specified later) and set
\[
L:=\log\log N,\qquad t:=\left\lfloor L-2\log L\right\rfloor,\qquad
\delta:=\frac{1}{L},\qquad y:=L^2,\qquad x:=N^{1/t}.
\]
For $N$ sufficiently large we have $t\ge t_0$, $y<x$, and for every prime $p>y$ we have $1/p\le 1/y\le\delta$. By Mertens' theorem,
\[
\sum_{p\le u}\frac1p=\log\log u+O(1)\qquad(u\to\infty),
\]
so
\begin{equation}\label{eq:enough-harmonic-mass}
\sum_{y<p\le x}\frac1p= \left(L- \log t\right) -\log\log y + O(1) \geq t(1+\delta)
\end{equation}for $N$ sufficiently large.

We next apply \eqref{eq:sathe-selberg-squarefree} with $x=t$. To ensure that the uniformity range for $\ell$ is satisfied, we restrict the
integral to a suitable subinterval where $\log\log t$ is comparable to $L$.
Set \(I := \left\{t\in[2,N] : \tfrac12 L \le \log\log t \le L \right\}\). Thus, for $t\in I$,
\[
\ell \le (1-\eta)L \le 2(1-\eta)\log\log t.
\]
Choose $\varepsilon=2\eta$, so that $2(1-\eta) = 2-\varepsilon$. Then for
all $t\in I$ and all $2\le \ell\le (1-\eta)L$ we have
\[
1\le \ell\le (2-\varepsilon)\log\log t.
\]
Hence \eqref{eq:sathe-selberg-squarefree} applies to $A_\ell(t)$ with $x=t$ for all $t\in I$, giving
\begin{equation}\label{eq:A-ell-asymp}
A_\ell(t)
= G\left(\frac{\ell-1}{\log\log t}\right)
 \frac{t}{\log t}
 \frac{(\log\log t)^{\ell-1}}{(\ell-1)!}
 \left(1 + O_{\eta}\left(\frac{\ell}{(\log\log t)^2}\right)\right)
\end{equation}
for all $t\in I$. Substituting \eqref{eq:A-ell-asymp} into \eqref{eq:H-ell-int} and restricting the integral to $t\in I$, we obtain
\begin{align*}
H_\ell(N)
 &\ge \int_{t\in I} \frac{A_\ell(t)}{t^2} dt\\
&= \frac1{(\ell-1)!}\int_{t\in I}
 G\left(\frac{\ell-1}{\log\log t}\right)
 (\log\log t)^{\ell-1}
 \frac{dt}{t\log t}
 \left(1 + O_{\eta}\left(\frac{\ell}{(\log\log t)^2}\right)\right).
\end{align*}For $t\in I$ we have $\log\log t\asymp L$, and since
$\ell\le (1-\eta)L$, we have
\[
\frac{\ell}{(\log\log t)^2}
\ll_\eta \frac1L.
\]
Thus, for $N$ sufficiently large, the factor
$1 + O_{\eta}\bigl(\ell/(\log\log t)^2\bigr)$ is bounded below by, say, $1/2$ on~$I$.
Moreover, for $t\in I$ we have
\[
0 \le \frac{\ell-1}{\log\log t}
\le \frac{(1-\eta)L}{L/2}
= 2(1-\eta) < 2,
\]
so that the argument of $G$ stays in the compact interval
$[0,2(1-\eta)]\subseteq[0,2)$. By continuity and positivity of $G$ on $[0,2)$,
there exists a constant $c_\eta>0$ such that
\[
G(z) \ge c_\eta \qquad\text{for all } 0\le z\le 2(1-\eta).
\]Therefore, using the substitution $u=\log\log t$ (so $du = dt/(t\log t)$), we
obtain
\begin{align*}
H_\ell(N)
&\ge \frac{1}{2(\ell-1)!} \int_{t\in I}
 G\left(\frac{\ell-1}{\log\log t}\right)
 (\log\log t)^{\ell-1}
 \frac{dt}{t\log t}
\\&\ge
\frac{c_\eta}{2(\ell-1)!}
\int_{\frac12 L}^{L} u^{\ell-1} du.
\end{align*}
The last integral can be evaluated explicitly:
\[
\int_{\frac12 L}^{L} u^{\ell-1} du
= \frac{L^\ell}{\ell}\left(1 - 2^{-\ell}\right)
\ge \frac{L^\ell}{2\ell}.
\]Hence
\[
H_\ell(N)
\ge
\frac{c_\eta}{2(\ell-1)!}\cdot \frac{L^\ell}{2\ell}
= \frac{c_\eta}{4} \frac{L^\ell}{\ell!}.
\]This shows that for all sufficiently large $N$ and all integers
$2\le \ell\le (1-\eta)L$ we have
\[
H_\ell(N) \ge C_\eta\,\frac{L^\ell}{\ell!}
\]
with $C_\eta = c_\eta/4>0$ depending only on~$\eta$. Together with the case
$\ell=1$ treated above, this proves that
\[
H_\ell(N) \gg_\eta \frac{L^\ell}{\ell!}
\]
uniformly for $1\le\ell\le(1-\eta)L$ as $N\to\infty$. This is~\eqref{eq:H-ell-lower}, and the lemma is proved.\qed

Our construction uses a ``prime bucketing'' scheme, which will be used for all lower bound constructions. We partition primes in a suitable interval
into many disjoint blocks of comparable harmonic sum and form squarefree integers by choosing
exactly $k-2$ primes from each block. A short combinatorial lemma forces LCM-$k$-freeness, while the
harmonic sum factorizes and can be optimized over a free parameter.

\subsection{Connections with the sunflower problem}
We next investigate the optimal exponent in the polylogarithmic bound.  Erd\H{o}s suggested that Problem~\ref{prob:856} is related to the sunflower problem. Recall that a collection of $k$ distinct sets $S_1,\dots,S_k$ is called a \emph{$k$-sunflower} if
\[
S_i\cap S_j = S_1\cap S_2 \qquad \text{for all }1\le i<j\le k.
\]
This common intersection is called the \emph{kernel}. A family of sets $\mathcal F$ is said to be
\emph{$k$-sunflower-free} if it contains no $k$ distinct members forming a $k$-sunflower. Dually, a collection of $k$ distinct sets $S_1,\dots,S_k$ is called a \emph{$k$-cosunflower} if \[
S_i\cup S_j = S_1\cup S_2 \qquad \text{for all }1\le i<j\le k.
\] A family $\mathcal F$ is \emph{$k$-cosunflower-free} if it
contains no $k$ distinct members forming a $k$-cosunflower.

In~\cite[p.~127]{Er70}, Erd\H{o}s wrote:
\begin{quote}
\emph{Problem~\ref{prob:856} seems connected with the following combinatorial problem:}
\end{quote}

\begin{prob}\label{prob:857}
Let $m=m(n,k)$ be minimal such that any distinct collection of sets
$A_1,\dots,A_m\subseteq [n]$ contains a $k$-sunflower. Estimate $m(n,k)$, or ideally determine
its asymptotic behavior.
\end{prob}

Problem~\ref{prob:857} is widely known as the Erd\H{o}s--Szemer\'edi sunflower conjecture~\cite{ErSz78},
and it appears as Problem~\#857 on Bloom's Erd\H{o}s Problems website~\cite{EP857}. Following the
notation of~\cite{NaSa17}, let $F_k(n)$ be the maximum size of a $k$-sunflower-free family
$\mathcal{F}\subseteq 2^{[n]}$, and define the \emph{Erd\H{o}s--Szemer\'edi $k$-sunflower-free capacity}
\[
\mu_k^{\mathrm S}:=\limsup_{n\to\infty} F_k(n)^{1/n}.
\]Trivially $\mu_k^{\mathrm S}\le 2$, and the celebrated Erd\H{o}s--Szemer\'edi sunflower conjecture~\cite{ErSz78} asserts that
$\mu_k^{\mathrm S}<2$ for every $k\ge 3$. A standard tensor power argument\footnote{Here is a concise description of the argument. Let $\mathcal{F}_n\subseteq 2^{[n]}$ be a $k$-sunflower-free family. Choose $r\in\{0,1,\dots,n\}$ maximizing $|\mathcal{F}_n\cap\binom{[n]}{r}|$, and set $\mathcal{F}_n':=\mathcal{F}_n\cap\binom{[n]}{r}$, so that $|\mathcal{F}_n'|\ge \frac{1}{n+1}|\mathcal{F}_n|$. For each integer $t\ge 2$, partition $[tn]$ into $t$ subsets $S_1\sqcup\cdots\sqcup S_t$ with $|S_i|=n$. For each $i$ fix a bijection $\pi_i:[n]\to S_i$ and define a copy of $\mathcal{F}_n'$ on $S_i$ by $\mathcal{F}_n'^{(i)}:=\{\pi_i(F):F\in\mathcal{F}_n'\}\subseteq 2^{S_i}$. Then one checks that $\mathcal{F}_{n}^{\otimes t}:=\{F_1\sqcup\cdots\sqcup F_t:\ F_i\in\mathcal{F}_n'^{(i)}\}\subseteq 2^{[tn]}$ is $k$-sunflower-free and satisfies $|\mathcal{F}_{n}^{\otimes t}|=|\mathcal{F}_n'|^t\ge (|\mathcal{F}_n|/(n+1))^t$.} shows that the limsup can be replaced with a limit:
\begin{equation}
    \label{eq:lim}
  \mu_k^{\mathrm S} =\lim_{n\to\infty} F_k(n)^{1/n}.
\end{equation}

In this paper, we make Erd\H{o}s's suggested connection between Problem~\ref{prob:856} and Problem~\ref{prob:857} precise by showing the Erd\H{o}s--Szemer\'edi sunflower conjecture is equivalent to a tight bound on $f_k(N)$.

\begin{thm}\label{thm:equivalence}
For each $k\ge 3$, we have $\mu_k^{\mathrm S} = 2$ if and only if $f_k(N) = (\log N)^{1 - o(1)}$.
\end{thm}
The implication ``$\Rightarrow$'' is obtained by constructing a LCM-$k$-free set of squarefree integers
whose prime-support sets form a large cosunflower-free family in an appropriate weighted
product measure. The implication ``$\Leftarrow$'' follows from Theorem~\ref{thm:B-implies-A} below.
\begin{thm}\label{thm:B-implies-A}
Fix $k\ge 3$. Then, as $N\to\infty$,
\[
f_k(N)\ll (\log N)^{\mu_k^{\mathrm S}-1+o(1)}.
\]
\end{thm}
Its proof is a double-counting argument similar to Erd\H{o}s's original argument in~\cite{Er70}. To each integer $m$ we
associate a set system on the prime divisors of $m$ and show that LCM-$k$-freeness forces these set
systems to be $k$-sunflower-free. We then combine the resulting estimate with a Sathe--Selberg type
lower bound for the harmonic sum of squarefree almost primes.

Finally, we wonder how good a lower bound for $f_k(N)$ we can establish given knowledge of $\mu_k^{\mathrm S}$. While we can obtain a lower bound by following the proof of Theorem~\ref{thm:equivalence}, it is not good when $\mu_k^{\mathrm S}$ is smaller than $2$. Towards an improvement on Theorem~\ref{thm:lower}, we show that
\begin{thm}\label{thm:mu-lb-implies-f-lb}
Fix $k\ge 3$. Then, as $N\to\infty$,
\[f_k(N) \ge (\log N)^{\log\mu_k^{\mathrm S} - o(1)}.\]
\end{thm}
For $k=3$, the best current upper bound due to Naslund and Sawin~\cite[Theorem~1]{NaSa17} states that
\[
\mu_3^{\mathrm S}\le \frac{3}{2^{2/3}}=1.889881574\dots\, .
\]
In the opposite direction, a construction of Deuber, Erd\H{o}s, Gunderson, Kostochka and
Meyer~\cite{DEGKM97} yields \(\mu_3^{\mathrm S}>1.551\). Hence, by Theorems~\ref{thm:B-implies-A} and~\ref{thm:mu-lb-implies-f-lb}, we obtain the following consequence.
\begin{cor}\label{cor:f3-upper}
As $N\to\infty$,
\[
(\log N)^{\log(1.551) - o(1)} \leq f_3(N)\ll (\log N)^{\frac{3}{2^{2/3}}-1+o(1)}.
\]
\end{cor}
Unfortunately, Theorem~\ref{thm:mu-lb-implies-f-lb} never gives an exponent greater than $\log 2 < 0.7$, so it is strictly worse than Theorem~\ref{thm:lower} for $k \geq 7$. We leave it as an open question to obtain the correct exponent.
\begin{prob}
    In terms of $\mu_k^{\mathrm S}$, determine the smallest $c > 0$ such that for all sufficiently large $N$, we have
    $$f_k(N) \leq (\log N)^{c + o(1)}.$$
\end{prob}

The rest of the paper is organized as follows.
In Section~\ref{sec:preliminaries} we fix notation and record two analytic inputs.
In Section~\ref{sec:lower} we prove our unconditional lower bound
(Theorem~\ref{thm:lower}). In Section~\ref{sec:upper} we prove the capacity-based upper bound (Theorem~\ref{thm:B-implies-A}).
In Section~\ref{sec:equivalence} we prove the full-density equivalence in
Theorem~\ref{thm:equivalence}. In Section~\ref{sec:lower-bound-via-capacity} we establish the capacity-based lower bound Theorem~\ref{thm:mu-lb-implies-f-lb}.
Finally, the appendix contains the proof of the technical Lemma~\ref{lem:harmonic-SS}.

To prove Theorem~\ref{thm:lower}, they use a ``prime bucketing'' construction: partition primes in a suitable interval into many disjoint blocks of comparable harmonic sum, and form squarefree integers by choosing exactly $k-2$ primes from each block. A short combinatorial lemma then forces LCM-$k$-freeness, while the harmonic sum factorizes and can be optimized over a free parameter.

To prove the main theorem, they use a ``prime bucketing'' construction: partition primes in a suitable interval into many disjoint blocks of comparable harmonic sum, and form squarefree integers by choosing exactly $k-2$ primes from each block. A short combinatorial lemma then forces LCM-$k$-freeness, while the harmonic sum factorizes and can be optimized over a free parameter.,

\label{thm:lower}
Fix an integer $k\ge 3$. Then, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{c_k-o(1)},
\qquad
c_k:=\frac{k-2}{e ((k-2)!)^{1/(k-2)}}.
\],

Question:
Fix an integer $k\ge 3$. For $N\ge 1$, let $[N]=\{1,2,\dots,N\}$. A set $A\subseteq [N]$ is called LCM-$k$-free if it contains no $k$ distinct elements $a_1,\dots,a_k\in A$ such that all pairwise least common multiples are equal, i.e. $\operatorname{lcm}(a_i,a_j)$ has the same value for every $1\le i<j\le k$. Define
\[
f_k(N):=\max\left\{\sum_{a\in A}\frac1a:\ A\subseteq [N]\text{ is LCM-}k\text{-free}\right\}.
\]
Which quantitative lower bound holds for $f_k(N)$ as $N\to\infty$?

For every fixed integer $k\ge 3$, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{c_k-o(1)},
\qquad c_k:=\frac{k-2}{e\,((k-2)!)^{1/(k-2)}}.
\]

Choices:
B.
For every fixed integer $k\ge 3$, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{1-o(1)}.
\]
C.
For every fixed integer $k\ge 3$, there exists a constant $c_k'>0$ such that as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{c_k'-o(1)}.
\]
D.
For every fixed integer $k\ge 3$, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{c_k+o(1)},
\qquad c_k:=\frac{k-2}{e\,((k-2)!)^{1/(k-2)}}.
\]
E.
For every fixed integer $k\ge 3$, as $N\to\infty$,
\[
f_k(N)\ge (\log N)^{d_k-o(1)},
\qquad d_k:=\frac{k-1}{e\,((k-1)!)^{1/(k-1)}}.
\]

optimization-over-free-parameter only yields a specific exponent strictly below 1

explicit formula for the exponent constant

sign of the little-o term in the exponent

block size is exactly $k-2$, so replacing it by $k-1$ alters the optimized exponent


\label{t:sup_sup}
  Assume \eqref{eq:degen} and \eqref{eq:powerlike}. Let $u$ be a strong solution to \eqref{eq:pde}--\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\widetilde u$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a supersolution and
  \begin{equation}
    \label{eq:sup_sup_n}
    u(x,t)
    \le
    \widetilde u(x,t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sup_sup_nn}
    \supp u(t)
    \subset
    B_{R(t)}(0)
    \,,
    \quad
    R(t)
    =
    \gamma
    g^{(-1)}(\log t)
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  for a suitable constant $\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\sup_{\R^{\SpDim}} u_{0}$, $R(0)$.

\begin{remark}
  \label{r:rad_N}
  Clearly the sup estimate valid for $v$ as in Theorems
  \ref{t:sup_sup_dens} and \ref{t:sub_sub_dens} is the same as in
  \eqref{eq:sup_decay}, which should be compared with the result valid for $\rw(s)\sim s^{-\alpha}$, $\alpha>p$, that is the universal bound $t^{-1/(p+m-3)}$ (see \cite{Andreucci:Tedeev:2021b}). Our present bound however depends on the initial data, as the known bound valid in the subcritical case $\alpha<p$, whose decay rate is $t^{-(N-\alpha)/K}$, $K=(N-\alpha)(p+m-3)+p-\alpha$.
\end{remark}

\begin{split}
      \ups(x,0)
      &\ge
        C_{*}
        t_{0}^{-\frac{1}{p+m-3}}
        \Big[
        \tar(\varGamma \log t_{0})^{\frac{1}{p-1}}
        -
        \rat(\rpol_{0})^{\frac{1}{p-1}}
        \Big]^{\frac{p-1}{p+m-3}}
        \\
      &\ge
        C_{*}
        t_{0}^{-\frac{1}{p+m-3}}
        \Big[
        (\varGamma \log t_{0})^{\frac{p-\alpha_{2}}{\alpha_{2}(p-1)}}
        -
        \rat(\rpol_{0})^{\frac{1}{p-1}}
        \Big]^{\frac{p-1}{p+m-3}}
        \ge
        M
        \,,
    \end{split}
  \end{equation}
  where last inequality is guaranteed by a suitable choice of $\varGamma$, of course preserving \eqref{eq:sss_ups_Gamma0}.
\end{proof}

\section{Self similar subsolutions}
\label{s:ssb}

We look for subsolutions in the radial form
\eqref{eq:sss_sup}--\eqref{eq:sss_ups_2}. We use here the same
notation introduced in Section~\ref{s:sss}, and we look at the set
where $\ups>0$. In this Section, the choice of the constants is more
delicate; we find it convenient to establish it from the beginning. In
this connection, let us note that the constant $\tilde\mu_{1}>0$ is
defined in \eqref{eq:sss_ubs_mu1}, $\tilde\mu_{2}>0$ in
\eqref{eq:sss_ubs_mu2}, and $\tilde\mu_{3}\ge 0$ in
\eqref{eq:sss_ubs_mu3}; the $\tilde\mu_{i}$ depend on $p$, $m$,
$\alpha_{1}$, $\alpha_{2}$ only. The constant $\nu_{0}$ is chosen as
in \eqref{eq:sss_ups_c1}. We also introduce for future reference a
given constant $\lambda>0$.
\\
We fix $t_{0}>1$ such that
\begin{equation}
  \label{eq:sss_ubs_t0}
  \begin{split}
    \log t_{0}
    \ge
      \max&\Big\{
      4
      \frac{p-\alpha_{1}}{\alpha_{1}}
      \,,
      \tilde\mu_{4}^{-1}
      \,,
      \frac{2\tilde\mu_{1}(N-1)+\tilde\mu_{3}
      +
      2\tilde\mu_{1}
      \alpha_{2}^{2}}
      {\tilde\mu_{4}}
      \,,
    \\
    &\quad
      \frac{2(\nu_{0}p)^{p-1}}{\tilde\mu_{4}(p+m-3)^{p-2}}
      (
      N
      +
      \alpha_{2}^{2}
      )
      \Big\}
      \,,
  \end{split}

\begin{equation}
  \label{eq:sup_decay}
  u(x,t)
  \le
  \gamma
  \Big[
  \frac{\exw^{(-1)}(\log t)^{p}}{t\log t}
  \Big]^{\frac{1}{p+m-3}}
  \,,
  \quad
  x\in\RN
  \,,
  t\gg 1
  \,,
\end{equation}

\begin{equation}
  \label{eq:fsp}
  \bar R(t)
  \le
  \gamma
  \exw^{(-1)}(\log t)
  \,,
  \quad
  t\gg 1
  \,,
\end{equation}

\begin{equation}
  \label{eq:powerlike}
  \alpha_{1}
  \frac{\exw(s)}{s}
  \le
  \exw'(s)
  \le
  \alpha_{2}
  \frac{\exw(s)}{s}
  \,,
  \qquad
  s>0
  \,.
\end{equation}

\begin{lemma}
  \label{l:sss_excf}
  We have
  \begin{equation}
    \label{eq:sss_excf_n}
    \frac{\exw(\rpol)}{\alpha_{2}}
    \le
    \excf(\rpol)
    \le
    \frac{\exw(\rpol)}{\alpha_{1}}
    \,,
    \qquad
    \rpol>0
    \,,
  \end{equation}
  and its obvious consequence
  \begin{equation}
    \label{eq:sss_excf_nn}
    \alpha_{1}
    \frac{\excf(\rpol)}{\rpol}
    \le
    \excf'(\rpol)
    =
    \frac{\exw(\rpol)}{\rpol}
    \le
    \alpha_{2}
    \frac{\excf(\rpol)}{\rpol}
    \,,
    \qquad
    \rpol>0
    \,.
  \end{equation}
\end{lemma}

\begin{theorem}
  \label{t:transform}
Radial solutions $U(\rpol,t)$ [respectively, supersolutions and subsolutions] of \eqref{eq:pde} correspond, thru a suitable transformation of the space variable, to radial solutions $v(s,t)$ [respectively, supersolutions and subsolutions] of \eqref{eq:rad_N},
for a suitable positive function $\rw\in C([0,+\infty))$ such that $\rw(0)=1$ and
\begin{equation}
  \label{eq:rad_rw_est}
  \rw(s)
  \sim
  \frac{\exw^{(-1)}(\log s)^{p}}{(\log s)^{p}}
  \frac{1}{s^{p}}
  \,.
\end{equation}
In fact, the transformation $r=\rpolf(s)$, $s>0$, is such that
\begin{equation}
  \label{eq:rad_r_asy_n}
  \rpolf(s)
  \sim
  \exw^{(-1)}(\log s)
  \,.
\end{equation}
\end{theorem}

\label{s:pre}

Note that as a consequence of \eqref{eq:powerlike} we have
\begin{equation}
  \label{eq:powerlike_inv}
  \alpha_{2}^{-1}
  \frac{\exw^{(-1)}(t)}{t}
  \le
  \der{\exw^{(-1)}}{t}(t)
  \le

\begin{equation}
    \label{eq:weaksol_a}
    \int_{0}^{+\infty}
    \int_{\RN}
    \{
    -
    u
    \eta_{t}
    +
    u^{m-1}
    \abs{\grad u}^{p-2}
    \grad u
    \grad \eta
    \}
    \ew(x)
    \di x
    \di t
    =
    0
    \,.
  \end{equation}

\begin{alignat}{2}
  \label{eq:pde}
  \ew(x)
  \pder{u}{t}
  -
  \Div\big(
  \ew(x)
  u^{m-1}
  \abs{\grad u}^{p-2}
  \grad u
  \big)
  &=
    0
    \,,
  &\qquad&
           \text{in $S_{T}$,}
  \\
  \label{eq:initd}
  u(x,0)
  &=
    u_{0}(x)
    \,,
  &\qquad&
           x\in\RN
           \,.
\end{alignat}

\begin{gather}
  \label{eq:powerlike_gt}
  \lambda^{\alpha_{1}}
  \exw(\rpol)
  \le
  \exw(\lambda\rpol)
  \le
  \lambda^{\alpha_{2}}
  \exw(\rpol)
  \,,
  \qquad
  \lambda\ge 1
  \,,
  \\
  \label{eq:powerlike_lt}
  \lambda^{\alpha_{2}}
  \exw(\rpol)
  \le
  \exw(\lambda\rpol)
  \le
  \lambda^{\alpha_{1}}
  \exw(\rpol)
  \,,
  \qquad
  \lambda\le 1
  \,.
\end{gather}

\begin{equation}
  \label{eq:sss_powerlike_gt_excf}
  \lambda^{\frac{1}{\alpha_{2}}}
  \excf^{(-1)}(\rpol)
  \le
  \excf^{(-1)}(\lambda\rpol)
  \le
  \lambda^{\frac{1}{\alpha_{1}}}
  \excf^{(-1)}(\rpol)
  \,,
  \qquad
  \lambda\ge 1
  \,.
\end{equation}

\begin{theorem}
  \label{t:sup_sup}
  Assume \eqref{eq:degen} and \eqref{eq:powerlike}. Let $u$ be a strong solution to \eqref{eq:pde}--\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\ups$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a supersolution and
  \begin{equation}
    \label{eq:sup_sup_n}
    u(x,t)
    \le
    \ups(x,t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sup_sup_nn}
    \supp u(t)
    \subset
    B_{R(t)}(0)
    \,,
    \quad
    R(t)
    =
    \gamma
    \exw^{(-1)}(\log t)
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  for a suitable constant $\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\sup_{\RN} u_{0}$, $R(0)$.
\end{theorem}

\begin{split}
      \auxf'(\rpol)
      &=
        \rat(\rpol)^{\frac{1}{p-1}-1}
        \frac{\rat'(\rpol)}{p-1}
        \Big(
        \delta_{1}
        +
        \frac{\delta_{2}}{\excf(\rpol)}
        \Big)
        -
        \delta_{2}
        \rat(\rpol)^{\frac{1}{p-1}}
        \frac{\excf'(\rpol)}{\excf(\rpol)^{2}}
        \\
      &=
        \delta_{1}
        \rat(\rpol)^{\frac{1}{p-1}-1}
        \frac{\rat'(\rpol)}{p-1}
        +
        \delta_{2}
        \frac{\rat(\rpol)^{\frac{1}{p-1}}}{\rpol\excf(\rpol)}
        \Big[
        \frac{\rpol}{p-1}
        \frac{\rat'(\rpol)}{\rat(\rpol)}
        -
        \frac{\excf'(\rpol)\rpol}{\excf(\rpol)}
        \Big]
        \\
      &=
        \delta_{1}
        \rat(\rpol)^{\frac{1}{p-1}-1}
        \frac{\rat'(\rpol)}{p-1}
        +
        \frac{p}{p-1}
        \delta_{2}
        \frac{\rat(\rpol)^{\frac{1}{p-1}}}{\rpol\excf(\rpol)}
        \Big[
        1
        -
        \frac{\excf'(\rpol)\rpol}{\excf(\rpol)}
        \Big]
        \\
      &\ge
        \frac{\rat(\rpol)^{\frac{1}{p-1}}}{\rpol}
        \Big\{
        \delta_{1}
        \frac{p-\alpha_{2}}{p-1}
        +
        \frac{p}{p-1}
        \frac{\delta_{2}}{\excf(\rpol)}
        \Big[
        1
        -
        \frac{\excf'(\rpol)\rpol}{\excf(\rpol)}
        \Big]
        \Big\}
        \,.
    \end{split}
  \end{equation}
  The claim follows from the inequality $\auxf'(\rpol)\ge 0$, which can be proved, for $\rpol$ as in \eqref{eq:sss_comb_n}, on applying next \eqref{eq:sss_excf_n}.
\end{proof}

\section{Self similar supersolutions}
\label{s:sss}

We assume here $\varGamma\ge 1$ and
\begin{equation}
  \label{eq:sss_ups_t0r0}
  \excf(\rpol_{0})
  <
  \log t_{0}
  \le
  \varGamma
  \log t_{0}
  \,,
\end{equation}
so that the support of $\ups(t)$ contains the ball $\rpol\le\rpol_{0}$ for all $t\ge 0$; this follows from the fact that $\rat$, and therefore $\tar$, is an increasing function as proved in Lemma~\ref{l:sss_rat}. By the same token, the support of $\ups(t)$ is the ball of radius $\excf^{(-1)}(\tau)$.

For the reader's convenience we recall that we assume $\alpha_{2}<p$, and note that $\rpol_{0}$ will be chosen as in \eqref{eq:sss_ups_r0}, $C_{*}$ as in \eqref{eq:sss_ups_C}, $\nu_{0}$ as in \eqref{eq:sss_ups_c1}, $\varGamma$ as in \eqref{eq:sss_ups_Gamma0}, while $t_{0}$ is required to satisfy \eqref{eq:sss_ups_t0r0} and \eqref{eq:sss_ups_ineqs5}.

1) Case $\rpol\ge \rpol_{0}$.
With the notation above, we write
\begin{equation}
  \label{eq:sss_sup_2}
  \ups(x,t)
  =
  \frac{C_{*}\brt(\rpol,\tau)^{\frac{p-1}{p+m-3}}}{(t+t_{0})^{\frac{1}{p+m-3}}}
  \,,
  \quad
  \brt(\rpol,\tau)
  :=
  \big[
  \tar(\tau)^{\frac{1}{p-1}}
  -
  \rat(\rpol)^{\frac{1}{p-1}}
  \big]_{+}
  \,.
\end{equation}
Clearly we may restrict in our calculations to the open set where $\brt>0$. Thus we compute
\begin{equation}
  \label{eq:sss_ups_pow}
  \ups(x,t)^{m-1}
  =
  \frac{C_{*}^{m-1}\brt(\rpol,\tau)^{\frac{(m-1)(p-1)}{p+m-3}}}{(t+t_{0})^{\frac{m-1}{p+m-3}}}
  \,,
\end{equation}
and
\begin{equation}
  \label{eq:sss_ups_rder}
  \ups_{\rpol}(x,t)
  =
  -
  \frac{1}{p+m-3}
  \frac{C_{*}\brt(\rpol,\tau)^{\frac{p-1}{p+m-3}-1}}{(t+t_{0})^{\frac{1}{p+m-3}}}
  \rat(\rpol)^{\frac{1}{p-1}-1}
  \rat'(\rpol)
  \,.
\end{equation}
Thus, on recalling $\rat'(\rpol)\ge 0$, we obtain
\begin{equation}
  \label{eq:sss_ups_diff}
  \begin{split}
    I_{1}
    &:=
      \ups^{m-1}
      \abs{\ups_{\rpol}}^{p-2}
      \ups_{\rpol}
    \\
    &=
      -
      \frac{
      C_{*}^{p+m-2}
      }{
      (p+m-3)^{p-1}
      }
      \,
      \frac{
      \brt(\rpol,\tau)^{\frac{p-1}{p+m-3}}
      }{
      (t+t_{0})^{\frac{p+m-2}{p+m-3}}
      }
      \rat(\rpol)^{2-p}
      \rat'(\rpol)^{p-1}
      \,.
  \end{split}

\begin{theorem}
  \label{t:sup_sup_dens}
  Let $v$ be a strong solution to \eqref{eq:rad_N}, for the specific $\rw$ found in Theorem~\ref{t:transform}, under assumptions \eqref{eq:degen} and \eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported and bounded.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\ups(\rpolf(\abs{x}),t)$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a supersolution to \eqref{eq:rad_N}, and
  \begin{equation}
    \label{eq:sup_sup_dens_n}
    v(x,t)
    \le
    \ups(\rpolf(\abs{x}),t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sup_sup_dens_nn}
    \supp v(t)
    \subset
    B_{\tilde R(t)}(0)
    \,,
    \qquad
    \log \tilde R(t)
    \sim
    \log t
    \,,
    \qquad
    t>t_{0}
    \,.
  \end{equation}
  The constants involved in \eqref{eq:sup_sup_dens_n}--\eqref{eq:sup_sup_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.
\end{theorem}

\begin{equation}
  \label{eq:rad_N}
  \rw(\abs{x})
  \pder{v}{t}
  =
  \Div(
  v^{m-1}
  \abs{\grad v}^{p-2}
  \grad v
  )
  \,,
  \qquad
  x\in\R^{N}
  \,,
  t>0
  \,.
\end{equation}

\begin{split}
      \max_{x\in\RN}
      \ups(x,0)
      &=
        \ups(0,0)
        =
        \frac{C_{*}}{t_{0}^{\frac{1}{p+m-3}}}
        \Big[
        \tar(\varGamma \log t_{0})^{\frac{1}{p-1}}
        -
        (1-\nu_{0})
        \rat(\rpol_{0})^{\frac{1}{p-1}}
        \Big]^{\frac{p-1}{p+m-3}}_{+}
      \\
      &\le
        C_{*}
        \tar(\varGamma \log t_{0})^{p+m-3}
        =
        C_{*}
        \tar\big(2^{\frac{\alpha_{2}(p-1)}{p-\alpha_{2}}}\excf(\rpol_{0})\big)^{p+m-3}
      \\
      &\le
        \lambda\tar\big(2^{\frac{\alpha_{2}(p-1)}{p-\alpha_{2}}}\lambda^{p+m-3}\big)^{p+m-3}
        \,.
    \end{split}
  \end{equation}
  Clearly, by selecting a suitable $\lambda>0$, both \eqref{eq:ssb_comp_n} and \eqref{eq:ssb_comp_nn} are satisfied.
\end{proof}

\section{Transformation to an inhomogeneous density equation}
\label{s:rad}

In this Section we prove Theorem~\ref{t:transform}.
\\
Essentially, we want to move the weight out of the right hand side of \eqref{eq:pde_rad}.
To this end, we perform a change of the independent variable $\rpol$, introducing the function $\rpolf(s)$, $s>0$, and the new unknown $v(s,t)=U(\rpolf(s),t)$. Then clearly
\begin{equation*}
  U_{r}
  =
  v_{s}
  \rpolf_{s}^{-1}
  \,,
  \quad
  U^{m-1}
  \abs{U_{\rpol}}^{p-2}
  U_{\rpol}
  =
  v^{m-1}
  \abs{v_{s}}^{p-2}
  v_{s}
  \abs{\rpolf_{s}}^{-(p-2)}
  \rpolf_{s}^{-1}
  \,.
\end{equation*}
On recalling also
$\partial/\partial\rpol=\rpolf_{s}^{-1}\partial/\partial s$, and on
assuming $\rpolf_{s}>0$, we get that the right hand side of
\eqref{eq:pde_rad} equals
\begin{equation}
  \label{eq:div_new}
  \pder{}{s}
  \big[
  s^{N-1}
  v^{m-1}
  \abs{v_{s}}^{p-2}
  v_{s}
  Y(s)
  \big]
  \rpolf_{s}^{-1}
  \,,
\end{equation}
where
\begin{equation}
  \label{eq:div_coeff}
  Y(s)
  :=
  \ew(\rpolf(s))
  \frac{\rpolf(s)^{N-1}}{s^{N-1}}
  \rpolf_{s}(s)^{-(p-2)-1}
  \,,
  \qquad
  s>0
  \,.
\end{equation}
We choose here $\rpolf$ as a solution to $Y(s)=1$, that is more specifically to the problem
\begin{equation}
  \label{eq:rad_rpolf}
  \rpolf_{s}(s)
  =
  \Big(
  \ew(\rpolf(s))
  \frac{\rpolf(s)^{N-1}}{s^{N-1}}
  \Big)^{\frac{1}{p-1}}
  \,,
  \qquad
  \rpolf(1)
  =
  \rpol_{*}
  \,.
\end{equation}
We prove in Section~\ref{s:app} that for a suitable $\rpol_{*}>0$, depending only on $N$, $p$ and $\exw$, the solution $\rpolf(s)$ is defined and increasing over $(0,+\infty)$ and satisfies
\begin{equation*}
  \lim_{s\to 0+}
  \rpolf(s)
  =
  0
  \,,
  \qquad
  \lim_{s\to +\infty}
  \rpolf(s)
  =
  +\infty
  \,,
\end{equation*}
so that it can be extended to a continuous function over $[0,+\infty)$. Even more, we have $\rpolf\in C^{1}([0,+\infty))$, with $\rpolf_{s}(0)=1$.
\\
Then \eqref{eq:pde_rad} becomes
\begin{equation*}
  \rpolf^{N-1}
  \ew(\rpolf)
  \rpolf_{s}
  \pder{v}{t}
  =
  \pder{}{s}
  \big[
  s^{N-1}
  v^{m-1}
  \abs{v_{s}}^{p-2}
  v_{s}
  \big]
  \,,
\end{equation*}
that is in view of our choice $Y(s)=1$,
\begin{equation}
  \label{eq:rad_new}
  \rpolf_{s}^{p}
  \pder{v}{t}
  =
  \frac{1}{s^{N-1}}
  \pder{}{s}
  \big[
  s^{N-1}
  v^{m-1}
  \abs{v_{s}}^{p-2}
  v_{s}
  \big]
  \,.
\end{equation}
The right hand side of \eqref{eq:rad_new} can of course be understood as the divergence in the space variable $x$ such that $\abs{x}=s$; we use for the sake of simplicity the old variable names. Then if we define
\begin{equation}
  \label{eq:rad_rw}
  \rw(s)=\rpolf_{s}(s)^{p}
  \,,
\end{equation}
we have the equation \eqref{eq:rad_N}
for the radial function $v$. According to the results on $\rpolf$ outlined above, $\rw$ is a continuos positive function on $[0,+\infty)$.

Next we estimate the asymptotic behavior of $\rw(s)$. We denote by $C_{i}$ constants depending only on $N$, $p$, $\exw$, possibly varying from line to line. We get from \eqref{eq:rad_rpolf} that
\begin{equation*}
  \frac{1}{(\ew(\rpol)\rpol^{N-1})^{\frac{1}{p-1}}}
  =
  \frac{1}{\flopr(\rpol)^{\frac{N-1}{p-1}}}\flopr_{\rpol}(\rpol)
  \,,
\end{equation*}
where $\flopr=\rpolf^{(-1)}$.
Thus, on defining
\begin{equation*}
  \varphi(\rpol)
  :=
  \int_{\rpol}^{+\infty}
  \frac{\di z}{(\ew(z)z^{N-1})^{\frac{1}{p-1}}}
  \,,
\end{equation*}
we get
\begin{equation}
  \label{eq:rad_rs}
  \varphi(\rpol)
  =
  \int_{\rpol}^{+\infty}
  \frac{1}{\flopr(z)^{\frac{N-1}{p-1}}}\flopr_{z}(z)
  \di z
  =
  \frac{p-1}{N-p}
  \flopr(\rpol)^{-\frac{N-p}{p-1}}
  \,.
\end{equation}
Define the function
\begin{equation}
  \label{eq:rad_H}
  \Hf(r)
  =
  \Big[
  \frac{\exw(\rpol)}{\rpol}
  \big(
  \ew(\rpol)
  \rpol^{N-1}
  \big)^{\frac{1}{p-1}}
  \Big]^{-1}
  \,,
  \qquad
  \rpol>0
  \,.
\end{equation}
Let us calculate, recalling that $\ew(\rpol)=e^{\exw(\rpol)}$,
\begin{equation}
  \label{eq:rad_H_der}
  \begin{split}
    \Hf'(\rpol)
    &=
      -
      \Big[
      \frac{\exw(\rpol)}{\rpol}
      \big(
      \ew(\rpol)
      \rpol^{N-1}
      \big)^{\frac{1}{p-1}}
      \Big]^{-2}
      \big(
      \ew(\rpol)
      \rpol^{N-1}
      \big)^{\frac{1}{p-1}}
      \\
    &\quad
      \times\Big\{
      \Big[
      \frac{\exw'(\rpol)}{\rpol}
      -
      \frac{\exw(\rpol)}{\rpol^{2}}
      \Big]
      +
      \frac{\exw(\rpol)}{(p-1)\rpol}
      \big(
      \ew(\rpol)
      \rpol^{N-1}
      \big)^{-1}
  \\
    &\qquad
      \times\big[
      \exw'(\rpol)
      \ew(\rpol)
      \rpol^{N-1}
      +
      (N-1)
      \ew(\rpol)
      \rpol^{N-2}
      \big]
      \Big\}
      \\
    &=
      -
      \Big[
      \frac{\exw(\rpol)}{\rpol}
      \big(
      \ew(\rpol)
      \rpol^{N-1}
      \big)^{\frac{1}{p-1}}
      \Big]^{-1}
      \Big\{
      \Big[
      \frac{\exw'(\rpol)}{\exw(\rpol)}
      -
      \frac{1}{\rpol}
      \Big]
      \\
    &\quad
      +
      \frac{1}{p-1}
      \Big[
      \exw'(\rpol)
      +
      \frac{N-1}{\rpol}
      \Big]
      \Big\}
      \,.
  \end{split}

\label{s:intro}

We look at the Cauchy problem for the doubly degenerate weighted parabolic
equation
\begin{alignat}{2}
  \label{eq:pde}
  f(x)
  \pder{u}{t}
  -
  \Div\big(
  f(x)
  u^{m-1}
  \lvert\operatorname{\nabla} u\rvert^{p-2}
  \operatorname{\nabla} u
  \big)
  &=
    0
    \,,
  &\qquad&
           \text{in $S_{T}$,}
  \\
  \label{eq:initd}
  u(x,0)
  &=
    u_{0}(x)
    \,,
  &\qquad&
           x\in\R^{\SpDim}
           \,.
\end{alignat}
Here $S_{T}=\R^{\SpDim}\times (0,T)$, $0<T\le +\infty$, $x=(x_{1},\dots,x_{N})$, $\operatorname{\nabla} u$ [respectively, $\Div$] is the gradient [respectively, the divergence] with respect to $x$, and we denote
\begin{equation}
  \label{eq:expweight}
  f(x)
  =
  e^{g(\lvertx\rvert)}
  \,,
  \qquad
  x\in\R^{\SpDim}
  \,.
\end{equation}
We assume in this paper, even without further reference, that $u\ge 0$, $u_{0}\ge 0$, $u_{0}\in L^{\infty}(\R^{\SpDim})$ is compactly supported, and that
\begin{equation}
  \label{eq:degen}
  1<p<N
  \,,
  \qquad
  p+m-3>0
  \,.
\end{equation}
On $g$ we assume that $g\in C([0,+\infty])\cap C^{1}((0,+\infty))$ is such that $g(0)=0$, $g(s)>0$ for $s>0$, and for given $0<\alpha_{1}\le \alpha_{2}<p$
\begin{equation}
  \label{eq:powerlike}
  \alpha_{1}
  \frac{g(s)}{s}
  \le
  g'(s)
  \le
  \alpha_{2}
  \frac{g(s)}{s}
  \,,
  \qquad
  s>0
  \,.
\end{equation}

In \cite{Andreucci:Tedeev:2025} we obtained sup bounds for solutions
to \eqref{eq:pde}--\eqref{eq:initd} in the range $\alpha_{2}<1$, and
estimates of the support for all $\alpha_{2}<Np/(p-1)$. The approach
of \cite{Andreucci:Tedeev:2025} relies on suitable embedding
inequalities and iterative estimates, and is valid for radial or
compactly supported solutions; see also in this connection
\cite{Andreucci:Tedeev:2015} and \cite{Andreucci:Tedeev:2022b}. Here
we present instead new results both for the decay of the solutions in
the $L^{\infty}$ norm and for the finite speed of propagation, by
means of comparison with explicit supersolutions and subsolutions, in
the range $\alpha_{1}$, $\alpha_{2}\in (0,p)$. Our approach is
inspired by the paper \cite{Grillo:Muratori:Vazquez:2017}, where the
authors considered the porous media equation on manifolds. We treat
here the case of the doubly nonlinear equation, and, as a further
novelty even in the case of the porous media equation, we deal
with the case of non-power nonlinearities (satisfying
\eqref{eq:powerlike}). For example the Zygmund function
\begin{equation*}
  g(s)
  =
  s^{\alpha}
  [\log(s+c)]^{\beta}
  \,,
  \quad
  s\ge 0
  \,,
  \alpha>0
  \,,
  \beta>0
  \,,
  c>1
  \,,
\end{equation*}
falls in this class with $\alpha_{1}=\alpha$, $\alpha_{2}=\alpha+\beta$.
\\
We show here that the asymptotic rates found in \cite{Andreucci:Tedeev:2025} are indeed optimal. These bounds are given by
\begin{equation}
  \label{eq:sup_decay}
  u(x,t)
  \le
  \gamma
  \Big[
  \frac{g^{(-1)}(\log t)^{p}}{t\log t}
  \Big]^{\frac{1}{p+m-3}}
  \,,
  \quad
  x\in\R^{\SpDim}
  \,,
  t\gg 1
  \,,
\end{equation}
and
\begin{equation}
  \label{eq:fsp}
  \bar R(t)
  \le
  \gamma
  g^{(-1)}(\log t)
  \,,
  \quad
  t\gg 1
  \,,
\end{equation}
where, for solutions whose support is bounded for $t\ge 0$ we define
\begin{equation}
  \label{eq:main_R}
  \bar R(t)
  =
  \inf\{
  r>0
  \mid
  \supp u(t) \subset B_{r}(0)
  \}
  \,.
\end{equation}
Here $\gamma>0$ is a constant depending on $u_{0}$.
According to our present results, the estimates in \eqref{eq:sup_decay}, \eqref{eq:fsp} are indeed sharp, at least for $\alpha_{2}<p$, and, even more, they are extended to such a range for bounded and compactly supported initial data. In addition the comparison results obtained here yield more precise pointwise estimates, tracking the dependence on $\lvertx\rvert$, in comparison to the bounds for the $L^{\infty}(\R^{\SpDim})$ obtained in the quoted papers of the present authors. This is an obvious effect of the barrier function method, which however is in principle less general than the approach through integral inequalities.

As in \cite{Grillo:Muratori:Vazquez:2017}, we may employ our barrier functions to investigate an equation with inhomogeneous density, i.e.,
\begin{equation}
  \label{eq:rad_N}
  \rho(\lvertx\rvert)
  \pder{v}{t}
  =
  \Div(
  v^{m-1}
  \lvert\operatorname{\nabla} v\rvert^{p-2}
  \operatorname{\nabla} v
  )
  \,,
  \qquad
  x\in\numberspacefont{R}^{N}
  \,,
  t>0
  \,.
\end{equation}
For the concept of solutions of \eqref{eq:rad_N}, and other related
information, we refer the reader to \cite{Andreucci:Tedeev:2021b}. In that paper, in fact in a more general setting, we considered equations modeled after \eqref{eq:rad_N}, with, roughly speaking, $\rho(s)$ behaving like $s^{-\alpha}$, $\alpha<p$ for large $s>0$. Here we deal with the new case \eqref{eq:rad_rw_est}, in which essentially $\rho(s)$ behaves like the power $s^{-p}$ multiplied by a powerlike function of $\log s$; this factor may be increasing or decreasing in $s$ according to the values of $\alpha_{1}$, $\alpha_{2}$. We refer to Remark~\ref{r:rad_N} for further comments on this point.

The Cauchy problem for nonlinear, and linear, parabolic equations
involving coefficients strongly depending on the space variable has
received great attention in the literature. We quote
\cite{Grigoryan:2006} which analyzes stochastical completeness and
other qualitative properties of weighted manifolds. In
\cite{Grillo:Muratori:2016} the authors study the smoothness and
temporal decay estimates of the solution to the Cauchy problem for the
porous media equation on Cartan-Hadamard manifolds supporting
Poincar\'{e} inequality. See also \cite{Grillo:Muratori:Vazquez:2017}
for precise two-sides space-time estimates of the solution to the  same problem,
in the setting of
various classes of Cartan-Hadamard manifolds,
\cite{Muratori:2021} for a survey of results, and again
\cite{Muratori:Roncoroni:2022}
for new Sobolev type inequalities as well as sharp bounds of solutions.
We also quote \cite{Tedeev:2007}, dealing with the influence of an inhomogeneous density on
the interface blow up phenomenon, on finite speed of
propagation, and in general on the
behavior of  the solution to the Cauchy problem for doubly degenerate
parabolic equations.
Finally
\cite{Tedeev:2025} investigates the blow up of the solution itself in the same setting.

\subsection{Main results}
\label{s:main}

We define here the spaces $L^{p}_{f}(\R^{\SpDim})$ in a standard way, as the $L^{p}$ spaces relative to the measure $f(x)\di x$.

\begin{definition}
  \label{d:weaksol}
  A function $u\ge 0$ is a weak solution to \eqref{eq:pde} if
  \begin{equation*}
    u\in C([0,+\infty); L^{1}_{f}(\R^{\SpDim})) \cap L^{\infty}(\R^{\SpDim}\times(\bar t,+\infty))
  \end{equation*}
  for all $\bar t>0$, and
  \begin{equation*}
    \operatorname{\nabla} u^{\frac{p+m-2}{p-1}}
    \in
    L^{p}_{\textup{loc}}
    (S_{\infty})
    \,.
  \end{equation*}
  In addition we require the standard integral formulation, i.e., for all $\eta\in C_{0}^{\infty}(S_{\infty})$,
  \begin{equation}
    \label{eq:weaksol_a}
    \int_{0}^{+\infty}
    \int_{\R^{\SpDim}}
    \{
    -
    u
    \eta_{t}
    +
    u^{m-1}
    \lvert\operatorname{\nabla} u\rvert^{p-2}
    \operatorname{\nabla} u
    \operatorname{\nabla} \eta
    \}
    f(x)
    \di x
    \di t
    =
    0
    \,.
  \end{equation}
  \\
  If we also have $u_{t}\in L^{\infty}_{\textup{loc}}(0,+\infty); L^{1}_{f\,\textup{loc}}(\R^{\SpDim}))$, then $u$ is a strong solution.
  \\
  Supersolutions [respectively, subsolutions] to \eqref{eq:pde} are defined similarly, with the difference that, for $\eta\ge0$, we require in \eqref{eq:weaksol_a} an inequality sign $\ge $ [respectively, $\le$].
  \\
  Solutions to the Cauchy problem \eqref{eq:pde}--\eqref{eq:initd} are solutions to \eqref{eq:pde} satisfying $u(0)=u_{0}$ in the sense of $C([0,+\infty); L^{1}_{f}(\R^{\SpDim}))$.
\end{definition}

Existence and comparison results for strong solutions to the problems which we consider here can be proved, at least for sufficiently regular initial data, according for example to the methods of \cite{Degtyarev:Tedeev:2012}, \cite{Tsutsumi:1988}; here we focus rather on estimates of such solutions.

When we consider radial solutions of the form $u(x,t)=U(r,t)$,
$r=\lvertx\rvert$, the equation \eqref{eq:pde} can be written as
\begin{equation}
  \label{eq:pde_rad}
  r^{N-1}
  f(r)
  \pder{U}{t}
  =
  \pder{}{r}
  \big[
  r^{N-1}
  f(r)
  U^{m-1}
  \abs{U_{r}}^{p-2}
  U_{r}
  \big]
  \,,
  \qquad
  r>0
  \,,
  t>0
  \,,
\end{equation}
which in turn can be rephrased as
\begin{equation}
  \label{eq:sss_rad}
  \begin{split}
    \pder{U}{t}
    &=
      \pder{}{r}
      \big[
      U^{m-1}
      \abs{U_{r}}^{p-2}
      U_{r}
      \big]
    \\
    &\quad+
      \Big(
      \frac{N-1}{r}
      +
      g'
      \Big)
      \big[
      U^{m-1}
      \abs{U_{r}}^{p-2}
      U_{r}
      \big]
      \,,
      \qquad
      r>0
      \,,
      t>0
      \,.
  \end{split}
\end{equation}
Of course the differential inequalities characterizing radial supersolutions and subsolutions can be rephrased similarly.
Specifically, we consider supersolutions and subsolutions of the form
\begin{alignat}2
  \label{eq:sss_sup}
  \widetilde u(x,t)
  &=
    \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
    \Big[
    E(\tau)^{\frac{1}{p-1}}
    -
    J(r)^{\frac{1}{p-1}}
    \Big]^{\frac{p-1}{p+m-3}}_{+}
    \,,
    &\quad&
            r\ge \rpol_{0}
            \,,
  \\
  \label{eq:sss_ups_2}
  \widetilde u(x,t)
  &=
    \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
    \Big[
    E(\tau)^{\frac{1}{p-1}}
    -
    I(r)
    \Big]^{\frac{p-1}{p+m-3}}_{+}
    \,,
    &\quad&
            r<\rpol_{0}
            \,,
\end{alignat}
where $r=\lvertx\rvert$, and
\begin{gather*}
  G(r)
  =
  \int_{0}^{r}
  \frac{g(s)}{s}
  \di s
  \,,
  \quad
  r\ge 0
  \,;
  \qquad
  J(r)
  =
  \frac{r^{p}}{G(r)}
  \,,
  \quad
  r>0
  \,.
  \\
  \tau
  =
  \varGamma
  \log(t+t_{0})
  \,,
  \quad
  t\ge 0
  \,;
  \\
  E(\tau)
  =
  J(G^{(-1)}(\tau))
  =
  \frac{G^{(-1)}(\tau)^{p}}{\tau}
  \,,
  \quad
  \tau>0
  \,;
  \\
  I(r)
  =
  \nu_{0}
  \Big(
  \frac{r^{p}}{G(\rpol_{0})}
  \Big)^{\frac{1}{p-1}}
  +
  (1-\nu_{0})
  J(\rpol_{0})^{\frac{1}{p-1}}
  \,,
  \quad
  r>0
  \,.
\end{gather*}
Note that the integral defining $G$ converges according to \eqref{eq:powerlike_lt}.

Here $C_{*}$, $\varGamma$, $t_{0}> 1$, $\rpol_0$ and $\nu_{0}$ are positive constants to be chosen. In particular, $\nu_{0}\in(0,1)$ will be selected so that $\widetilde u$ and $\ups_{r}$ are continuous even at $r=\rpol_{0}$; in fact, clearly, the continuity of $\widetilde u$ holds true for any $\nu_{0}\in(0,1)$.

We also remark that the radius $\tilde r(t)$ of the support of $\widetilde u(t)$ is characterized by $E(\tau)=J(\tilder(t))$, i.e., by $\tilder(t)=G^{(-1)}(\tau)$; according to Lemma~\ref{l:sss_excf} and \eqref{eq:sss_powerlike_gt_excf} this yields
\begin{equation}
  \label{eq:sup_sub_radius}
  \gamma_{1}
  g^{(-1)}(\log t)
  \le
  \tilde r(t)
  \le
  \gamma_{2}
  g^{(-1)}(\log t)
  \,,
  \qquad
  t>t_{0}
  \,,
\end{equation}
for two suitable constants $\gamma_{1}$, $\gamma_{2}>0$.

\begin{proposition}
  \label{p:main_sup_sub}
  The constants $C_{*}$, $\varGamma$, $t_{0}> 1$, $\rpol_0$ and
  $\nu_{0}$ can be selected so that $\widetilde u$ as defined in
  \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a strong supersolution to
  \eqref{eq:pde}. Alternatively, we may select such constants so that
  $\widetilde u$ is a strong subsolution to \eqref{eq:pde}.
\end{proposition}

We look at the Cauchy problem for the doubly degenerate weighted parabolic
equation
\begin{alignat}{2}
 \label{eq:pde}
 f(x)
 \pder{u}{t}
 -
 \Div\big(
 f(x)
 u^{m-1}
 \lvert\operatorname{\nabla} u\rvert^{p-2}
 \operatorname{\nabla} u
 \big)
 &=
 0
 \,,
 &\qquad&
 \text{in $S_{T}$,}
 \\
 \label{eq:initd}
 u(x,0)
 &=
 u_{0}(x)
 \,,
 &\qquad&
 x\in\R^{\SpDim}
 \,.
\end{alignat}
Here $S_{T}=\R^{\SpDim}\times (0,T)$, $0<T\le +\infty$, $x=(x_{1},\dots,x_{N})$, $\operatorname{\nabla} u$ [respectively, $\Div$] is the gradient [respectively, the divergence] with respect to $x$, and we denote
\begin{equation}
 \label{eq:expweight}
 f(x)
 =
 e^{g(\lvertx\rvert)}
 \,,
 \qquad
 x\in\R^{\SpDim}
 \,.
\end{equation}
We assume in this paper, even without further reference, that $u\ge 0$, $u_{0}\ge 0$, $u_{0}\in L^{\infty}(\R^{\SpDim})$ is compactly supported, and that
\begin{equation}
 \label{eq:degen}
 1<p<N
 \,,
 \qquad
 p+m-3>0
 \,.
\end{equation}
On $g$ we assume that $g\in C([0,+\infty])\cap C^{1}((0,+\infty))$ is such that $g(0)=0$, $g(s)>0$ for $s>0$, and for given $0<\alpha_{1}\le \alpha_{2}<p$
\begin{equation}
 \label{eq:powerlike}
 \alpha_{1}
 \frac{g(s)}{s}
 \le
 g'(s)
 \le
 \alpha_{2}
 \frac{g(s)}{s}
 \,,
 \qquad
 s>0
 \,.
\end{equation}

\begin{definition}
 \label{d:weaksol}
 A function $u\ge 0$ is a weak solution to \eqref{eq:pde} if
 \begin{equation*}
 u\in C([0,+\infty); L^{1}_{f}(\R^{\SpDim})) \cap L^{\infty}(\R^{\SpDim}\times(\bar t,+\infty))
 \end{equation*}
 for all $\bar t>0$, and
 \begin{equation*}
 \operatorname{\nabla} u^{\frac{p+m-2}{p-1}}
 \in
 L^{p}_{\textup{loc}}
 (S_{\infty})
 \,.
 \end{equation*}
 In addition we require the standard integral formulation, i.e., for all $\eta\in C_{0}^{\infty}(S_{\infty})$,
 \begin{equation}
 \label{eq:weaksol_a}
 \int_{0}^{+\infty}
 \int_{\R^{\SpDim}}
 \{
 -
 u
 \eta_{t}
 +
 u^{m-1}
 \lvert\operatorname{\nabla} u\rvert^{p-2}
 \operatorname{\nabla} u
 \operatorname{\nabla} \eta
 \}
 f(x)
 \di x
 \di t
 =
 0
 \,.
 \end{equation}
 \\
 If we also have $u_{t}\in L^{\infty}_{\textup{loc}}(0,+\infty); L^{1}_{f\,\textup{loc}}(\R^{\SpDim}))$, then $u$ is a strong solution.
 \\
 Supersolutions [respectively, subsolutions] to \eqref{eq:pde} are defined similarly, with the difference that, for $\eta\ge0$, we require in \eqref{eq:weaksol_a} an inequality sign $\ge $ [respectively, $\le$].
 \\
 Solutions to the Cauchy problem \eqref{eq:pde}--\eqref{eq:initd} are solutions to \eqref{eq:pde} satisfying $u(0)=u_{0}$ in the sense of $C([0,+\infty); L^{1}_{f}(\R^{\SpDim}))$.
\end{definition}

When we consider radial solutions of the form $u(x,t)=U(r,t)$,
$r=\lvertx\rvert$, the equation \eqref{eq:pde} can be written as
\begin{equation}
 \label{eq:pde_rad}
 r^{N-1}
 f(r)
 \pder{U}{t}
 =
 \pder{}{r}
 \big[
 r^{N-1}
 f(r)
 U^{m-1}
 \abs{U_{r}}^{p-2}
 U_{r}
 \big]
 \,,
 \qquad
 r>0
 \,,
 t>0
 \,,
\end{equation}
which in turn can be rephrased as
\begin{equation}
 \label{eq:sss_rad}
 \begin{split}
 \pder{U}{t}
 &=
 \pder{}{r}
 \big[
 U^{m-1}
 \abs{U_{r}}^{p-2}
 U_{r}
 \big]
 \\
 &\quad+
 \Big(
 \frac{N-1}{r}
 +
 g'
 \Big)
 \big[
 U^{m-1}
 \abs{U_{r}}^{p-2}
 U_{r}
 \big]
 \,,
 \qquad
 r>0
 \,,
 t>0
 \,.
 \end{split}
\end{equation}
Of course the differential inequalities characterizing radial supersolutions and subsolutions can be rephrased similarly.
Specifically, we consider supersolutions and subsolutions of the form
\begin{alignat}2
 \label{eq:sss_sup}
 \widetilde u(x,t)
 &=
 \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
 \Big[
 E(\tau)^{\frac{1}{p-1}}
 -
 J(r)^{\frac{1}{p-1}}
 \Big]^{\frac{p-1}{p+m-3}}_{+}
 \,,
 &\quad&
 r\ge \rpol_{0}
 \,,
 \\
 \label{eq:sss_ups_2}
 \widetilde u(x,t)
 &=
 \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
 \Big[
 E(\tau)^{\frac{1}{p-1}}
 -
 I(r)
 \Big]^{\frac{p-1}{p+m-3}}_{+}
 \,,
 &\quad&
 r<\rpol_{0}
 \,,
\end{alignat}
where $r=\lvertx\rvert$, and
\begin{gather*}
 G(r)
 =
 \int_{0}^{r}
 \frac{g(s)}{s}
 \di s
 \,,
 \quad
 r\ge 0
 \,;
 \qquad
 J(r)
 =
 \frac{r^{p}}{G(r)}
 \,,
 \quad
 r>0
 \,.
 \\
 \tau
 =
 \varGamma
 \log(t+t_{0})
 \,,
 \quad
 t\ge 0
 \,;
 \\
 E(\tau)
 =
 J(G^{(-1)}(\tau))
 =
 \frac{G^{(-1)}(\tau)^{p}}{\tau}
 \,,
 \quad
 \tau>0
 \,;
 \\
 I(r)
 =
 \nu_{0}
 \Big(
 \frac{r^{p}}{G(\rpol_{0})}
 \Big)^{\frac{1}{p-1}}
 +
 (1-\nu_{0})
 J(\rpol_{0})^{\frac{1}{p-1}}
 \,,
 \quad
 r>0
 \,.
\end{gather*}
Note that the integral defining $G$ converges according to \eqref{eq:powerlike_lt}.

We also remark that the radius $\tilde r(t)$ of the support of $\widetilde u(t)$ is characterized by $E(\tau)=J(\tilder(t))$, i.e., by $\tilder(t)=G^{(-1)}(\tau)$; according to Lemma~\ref{l:sss_excf} and \eqref{eq:sss_powerlike_gt_excf} this yields
\begin{equation}
 \label{eq:sup_sub_radius}
 \gamma_{1}
 g^{(-1)}(\log t)
 \le
 \tilde r(t)
 \le
 \gamma_{2}
 g^{(-1)}(\log t)
 \,,
 \qquad
 t>t_{0}
 \,,
\end{equation}
for two suitable constants $\gamma_{1}$, $\gamma_{2}>0$.

\begin{proposition}
 \label{p:main_sup_sub}
 The constants $C_{*}$, $\varGamma$, $t_{0}> 1$, $\rpol_0$ and
 $\nu_{0}$ can be selected so that $\widetilde u$ as defined in
 \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a strong supersolution to
 \eqref{eq:pde}. Alternatively, we may select such constants so that
 $\widetilde u$ is a strong subsolution to \eqref{eq:pde}.
\end{proposition}

\begin{theorem}
 \label{t:sub_sub_dens}
 Let $v$ be a strong solution to \eqref{eq:rad_N}, for the specific $\rw$ found in Theorem~\ref{t:transform}, under assumptions \eqref{eq:degen} and \eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported, bounded and
$v(x,0)\ge \eps>0$ for $\abs{x}\le\ell$, $\ell>0$.
 \\
 Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\ups(\rpolf(\abs{x}),t)$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a subsolution \eqref{eq:rad_N}, and
 \begin{equation}
 \label{eq:sub_sub_dens_n}
 v(x,t)
 \ge
 \ups(\rpolf(\abs{x}),t)
 \,,
 \qquad
 (x,t)\in S_{\infty}
 \,.
 \end{equation}
 We have also
 \begin{equation}
 \label{eq:sub_sub_dens_nn}
 \supp v(t)
 \supset
 B_{\tilde R(t)}(0)
 \,,
 \quad
 \log\tilde R(t)
 \sim
 \log t
 \,,
 \qquad
 t>t_{0}
 \,,
 \end{equation}
 The constants involved in \eqref{eq:sub_sub_dens_n}--\eqref{eq:sub_sub_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.
\end{theorem}

We work in the set $\trb>0$. Then we compute
\begin{equation}
 \label{eq:sss_ups_diffs}
 \begin{split}
 L_{1}
 &:=
 \ups^{m-1}
 \abs{\ups_{\rpol}}^{p-2}
 \ups_{\rpol}
 \\
 &=
 -
 \frac{
 C_{*}^{p+m-2}
 (\nu_{0}p)^{p-1}
 }{
 (p+m-3)^{p-1}
 }
 \,
 \frac{
 \trb(\rpol,\tau)^{\frac{p-1}{p+m-3}}
 }{
 (t+t_{0})^{\frac{p+m-2}{p+m-3}}
 }
 \frac{\rpol}{\excf(\rpol_{0})}
 \,.
 \end{split}
\end{equation}
We recall that, in order for $\ups$ to be a supersolution, we need to check (see also \eqref{eq:sss_ups_ineq})
\begin{equation}
 \label{eq:sss_ups_ineqs}
 \pder{\ups}{t}
 \ge
 \pder{L_{1}}{\rpol}
 +
 L_{1}
 \Big(
 \frac{N-1}{\rpol}
 +
 \exw'(\rpol)
 \Big)
 \,,
 \qquad
 \rpol<\rpol_{0}
 \,.
\end{equation}
The time derivative $\ups_{t}$ is given by \eqref{eq:sss_ups_tder},
when we formally substitute $\brt(\rpol,\tau)$ with $\trb(\rpol,\tau)$.
Next we calculate
\begin{equation}
 \label{eq:sss_ups_diffs2}
 \begin{split}
 \pder{L_{1}}{\rpol}
 &=
 \frac{
 C_{*}^{p+m-2}
 (\nu_{0}p)^{p-1}
 }{
 (p+m-3)^{p-1}
 }
 \,
 \frac{
 \trb(\rpol,\tau)^{\frac{p-1}{p+m-3}-1}
 }{
 (t+t_{0})^{\frac{p+m-2}{p+m-3}}
 \excf(\rpol_{0})
 }
 \\
 &\quad\times
 \Big[
 \frac{p\nu_{0}}{p+m-3}
 \frac{\rpol^{\frac{p}{p-1}}}{\excf(\rpol_{0})^{\frac{1}{p-1}}}
 -
 \trb(\rpol,\tau)
 \Big]
 \,.
 \end{split}
\end{equation}
Upon dividing both sides of \eqref{eq:sss_ups_ineqs} by
\begin{equation*}
 \frac{C_{*}}{p+m-3}
 \,
 \frac{\trb(\rpol,\tau)^{\frac{p-1}{p+m-3}-1}}{(t+t_{0})^{\frac{p+m-2}{p+m-3}}}
 \,,
\end{equation*}
we obtain the equivalent version of \eqref{eq:sss_ups_ineqs} (here $K_{0}$ is as in \eqref{eq:sss_ups_ineq2})
\begin{equation}
 \label{eq:sss_ups_ineqs2}
 \begin{split}
 K_{0}&=
 (p-1)
 (t+t_{0})
 \der{}{t}
 \big(
 \tar(\tau)^{\frac{1}{p-1}}
 \big)
 \ge
 \trb(\rpol,\tau)
 \\
 &\quad+
 \frac{C_{*}^{p+m-3}(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}
 \frac{1}{\excf(\rpol_{0})}
 \Big[
 \frac{p\nu_{0}}{p+m-3}
 \frac{\rpol^{\frac{p}{p-1}}}{\excf(\rpol_{0})^{\frac{1}{p-1}}}
 -
 \trb(\rpol,\tau)
 \Big]
 \\
 &\quad
 -
 \Big(
 \frac{N-1}{\rpol}
 +
 \exw'(\rpol)
 \Big)
 \frac{C_{*}^{p+m-3}(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}
 \frac{\trb(\rpol,\tau)\rpol}{\excf(\rpol_{0})}
 =:
 K_{3}
 \,.
 \end{split}
\end{equation}
On recalling \eqref{eq:sss_ups_k0}, we see that $K_{0}>0$, so that we need only show
\begin{equation}
 \label{eq:sss_ups_ineqs3}
 0
 \ge
 K_{3}
 \,.
\end{equation}
Clearly, this is implied by
\begin{equation}
 \label{eq:sss_ups_ineqs4}
 \begin{split}
 & \Big(
 \frac{(N-1)C_{*}^{p+m-3}(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}\excf(\rpol_{0})}
 -
 1
 \Big)
 \trb(\rpol,\tau)
 \ge
 \\
 &\quad
 \frac{C_{*}^{p+m-3}(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}
 \frac{1}{\excf(\rpol_{0})}
 \Big[
 \frac{p\nu_{0}}{p+m-3}
 \frac{\rpol^{\frac{p}{p-1}}}{\excf(\rpol_{0})^{\frac{1}{p-1}}}
 -
 \trb(\rpol,\tau)
 \Big]
 \,.
 \end{split}
\end{equation}
The left hand side of \eqref{eq:sss_ups_ineqs4} is nonnegative, owing to our choice of $C_{*}$ and to \eqref{eq:sss_ups_c1} (we use the second term in the $\max$ function in \eqref{eq:sss_ups_C} here). Thus we only need to show that the right hand side there is nonpositive for $\rpol<\rpol_{0}$. According to the definitions of $\tau$ and of $\trb(\rpol,\tau)$ this in turn follows from
\begin{equation}
 \label{eq:sss_ups_ineqs5}
 \begin{split}
 &\frac{p\nu_{0}}{p+m-3}
 \frac{\rpol_{0}^{\frac{p}{p-1}}}{\excf(\rpol_{0})^{\frac{1}{p-1}}}
 +
 \ram(\rpol_{0})
 \le
 \tar(\log t_{0})^{\frac{1}{p-1}}
 \\
 &\quad\le
 \tar(\varGamma \log t_{0})^{\frac{1}{p-1}}
 \le
 \tar(\tau)^{\frac{1}{p-1}}
 \,,
 \end{split}
\end{equation}
where the last two inequalities follow trivially from the increasing character of $\tar(\tau)$ and $\varGamma\ge 1$. But since $\rpol_{0}$ has been fixed, this amounts simply to choosing $t_{0}=t_{0}(\rpol_{0})$ so that both \eqref{eq:sss_ups_t0r0} and \eqref{eq:sss_ups_ineqs5} are satisfied; actually it is easily seen that \eqref{eq:sss_ups_t0r0} follows from \eqref{eq:sss_ups_ineqs5}. Note that $r_{0}$, $C_{*}$ and $t_{0}$ are independent of $\varGamma$.

We look for subsolutions in the radial form
\eqref{eq:sss_sup}--\eqref{eq:sss_ups_2}. We use here the same
notation introduced in Section~\ref{s:sss}, and we look at the set
where $\ups>0$. In this Section, the choice of the constants is more
delicate; we find it convenient to establish it from the beginning. In
this connection, let us note that the constant $\tilde\mu_{1}>0$ is
defined in \eqref{eq:sss_ubs_mu1}, $\tilde\mu_{2}>0$ in
\eqref{eq:sss_ubs_mu2}, and $\tilde\mu_{3}\ge 0$ in
\eqref{eq:sss_ubs_mu3}; the $\tilde\mu_{i}$ depend on $p$, $m$,
$\alpha_{1}$, $\alpha_{2}$ only. The constant $\nu_{0}$ is chosen as
in \eqref{eq:sss_ups_c1}. We also introduce for future reference a
given constant $\lambda>0$.
\\
We fix $t_{0}>1$ such that
\begin{equation}
 \label{eq:sss_ubs_t0}
 \begin{split}
 \log t_{0}
 \ge
 \max&\Big\{
 4
 \frac{p-\alpha_{1}}{\alpha_{1}}
 \,,
 \tilde\mu_{4}^{-1}
 \,,
 \frac{2\tilde\mu_{1}(N-1)+\tilde\mu_{3}
 +
 2\tilde\mu_{1}
 \alpha_{2}^{2}}
 {\tilde\mu_{4}}
 \,,
 \\
 &\quad
 \frac{2(\nu_{0}p)^{p-1}}{\tilde\mu_{4}(p+m-3)^{p-2}}
 (
 N
 +
 \alpha_{2}^{2}
 )
 \Big\}
 \,,
 \end{split}
\end{equation}
where
\begin{equation}
 \label{eq:ssb_ups_mu4}
 \tilde\mu_{4}
 =
 2^{-\frac{\alpha_{2}(p-1)}{p-\alpha_{2}}}
 \frac{\tilde\mu_{2}\alpha_{1}}{p-\alpha_{1}}
 >0
 \,.
\end{equation}
Next we select $\rpol_{0}>0$ such that
\begin{equation}
 \label{eq:ssb_ups_r0}
 \excf(r_{0})
 <
 \min
 \Big\{
 1
 \,,
 \lambda^{p+m-3}
 \,,
 \tilde\mu_{4}
 C_{*}^{p+m-3}
 \log t_{0}
 \Big\}
 \,,
\end{equation}
where we set
\begin{equation}
 \label{eq:ssb_ups_C}
 \begin{split}
 C_{*}^{-(p+m-3)}
 &=
 \max\Big\{
 \lambda^{-(p+m-3)}
 \,,
 2\frac{\tilde\mu_{1}(N-1)+\tilde\mu_{3}}{\excf(\rpol_{0})}
 +
 2\tilde\mu_{1}
 \alpha_{2}^{2}
 \,,
 \\
 &\quad
 \frac{2(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}
 \frac{1}{\excf(\rpol_{0})}
 (
 N
 +
 \alpha_{2}^{2}
 \excf(\rpol_{0})
 )
 \Big\}
 \,.
 \end{split}
\end{equation}
Clearly we have to show that \eqref{eq:ssb_ups_r0}, \eqref{eq:ssb_ups_C} are compatible, that is that
\begin{equation}
 \label{eq:ssb_ups_r0_i}
 \excf(\rpol_{0})C_{*}^{-(p+m-3)}
 <
 \tilde\mu_{4}
 \log t_{0}
\end{equation}
can be fulfilled by choosing a small enough $\rpol_{0}$. Indeed, also invoking $\excf(\rpol_{0})<\min\{1,\lambda^{p+m-3}\}$ which is certainly meaningful, we compute from \eqref{eq:ssb_ups_C}
\begin{equation}
 \label{eq:ssb_ups_r0_ii}
 \begin{split}
 \excf(\rpol_{0})C_{*}^{-(p+m-3)}
 &<
 \max\Big\{
 1
 \,,
 2\tilde\mu_{1}(N-1)+\tilde\mu_{3}
 +
 2\tilde\mu_{1}
 \alpha_{2}^{2}
 \,,
 \\
 &\quad
 \frac{2(\nu_{0}p)^{p-1}}{(p+m-3)^{p-2}}
 (
 N
 +
 \alpha_{2}^{2}
 )
 \Big\}
 \le
 \tilde\mu_{4}
 \log t_{0}
 \,,
 \end{split}
\end{equation}
according to our choice of $t_{0}$.
Finally we select
\begin{equation}
 \label{eq:ssb_ups_Gamma0}
 \varGamma
 =
 2^{\frac{\alpha_{2}(p-1)}{p-\alpha_{2}}}
 \frac{\excf(\rpol_{0})}{\log t_{0}}
 \,.
\end{equation}

\begin{theorem}
  \label{t:sub_sub}
  Assume \eqref{eq:degen} and \eqref{eq:powerlike}. Let $u$ be a strong solution to \eqref{eq:pde}--\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$, such that $u_{0}(x)\ge \varepsilon>0$ for $\lvertx\rvert\le\ell$, $\ell>0$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\widetilde u$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a subsolution and
  \begin{equation}
    \label{eq:sub_sub_n}
    u(x,t)
    \ge
    \widetilde u(x,t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sub_sub_nn}
    \supp u(t)
    \supset
    B_{R(t)}(0)
    \,,
    \quad
    R(t)
    =
    \gamma
    g^{(-1)}(\log t)
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  for a suitable constant $\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\varepsilon$, $\ell$.
\end{theorem}

\begin{proof}[Proofs of Theorems \ref{t:sup_sup} and \ref{t:sub_sub}]
  The proof follows straightforwardly from an application of the method given in the proof of \cite[Theorem~1]{Tsutsumi:1988}, when we take into account Proposition~\ref{p:main_sup_sub} and Lemmas \ref{l:sss_comp} and \ref{l:ssb_comp}.
\end{proof}

Next we apply our results to a different kind of equation, that is \eqref{eq:rad_N}; see also \cite{Vazquez:2015}, \cite{Grillo:Muratori:Vazquez:2017} for the case of the porous media equation.

\begin{theorem}
  \label{t:transform}
Radial solutions $U(r,t)$ [respectively, supersolutions and subsolutions] of \eqref{eq:pde} correspond, thru a suitable transformation of the space variable, to radial solutions $v(s,t)$ [respectively, supersolutions and subsolutions] of \eqref{eq:rad_N},
for a suitable positive function $\rho\in C([0,+\infty))$ such that $\rho(0)=1$ and
\begin{equation}
  \label{eq:rad_rw_est}
  \rho(s)
  \sim
  \frac{g^{(-1)}(\log s)^{p}}{(\log s)^{p}}
  \frac{1}{s^{p}}
  \,.
\end{equation}
In fact, the transformation $r=\hat\rpol(s)$, $s>0$, is such that
\begin{equation}
  \label{eq:rad_r_asy_n}
  \hat\rpol(s)
  \sim
  g^{(-1)}(\log s)
  \,.
\end{equation}
\end{theorem}

Here and in the following we use the notation, for positive functions $F_{1}$, $F_{2}$,
\begin{equation}
  \label{eq:rad_sim}
  F_{1}(s)\sim F_{2}(s)
  \quad
  \text{if and only if}
  \quad
  C_{1}
  \le
  \frac{F_{2}(s)}{F_{1}(s)}
  \le
  C_{2}
  \,,
  \quad
  s\ge \bar s
  \,,
\end{equation}
for suitable $C_{1}$, $C_{2}$, $\bar s>0$.

Owing to Theorem~\ref{t:transform}, supersolutions and subsolutions of the type \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} yield radial counterparts for equation \eqref{eq:rad_N}. Thus, from our  Theorems \ref{t:sup_sup}, \ref{t:sub_sub} the following comparison results follow immediately.

\begin{theorem}
  \label{t:sup_sup_dens}
  Let $v$ be a strong solution to \eqref{eq:rad_N}, for the specific $\rho$ found in Theorem~\ref{t:transform}, under assumptions \eqref{eq:degen} and \eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported and bounded.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\widetilde u(\hat\rpol(\lvertx\rvert),t)$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a supersolution to \eqref{eq:rad_N}, and
  \begin{equation}
    \label{eq:sup_sup_dens_n}
    v(x,t)
    \le
    \widetilde u(\hat\rpol(\lvertx\rvert),t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sup_sup_dens_nn}
    \supp v(t)
    \subset
    B_{\tilde R(t)}(0)
    \,,
    \qquad
    \log \tilde R(t)
    \sim
    \log t
    \,,
    \qquad
    t>t_{0}
    \,.
  \end{equation}
  The constants involved in \eqref{eq:sup_sup_dens_n}--\eqref{eq:sup_sup_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.
\end{theorem}

\begin{theorem}
  \label{t:sub_sub_dens}
  Let $v$ be a strong solution to \eqref{eq:rad_N}, for the specific $\rho$ found in Theorem~\ref{t:transform}, under assumptions \eqref{eq:degen} and \eqref{eq:powerlike}, such that $v(x,0)$ is compactly supported, bounded and
$v(x,0)\ge \varepsilon>0$ for $\lvertx\rvert\le\ell$, $\ell>0$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\widetilde u(\hat\rpol(\lvertx\rvert),t)$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a subsolution \eqref{eq:rad_N}, and
  \begin{equation}
    \label{eq:sub_sub_dens_n}
    v(x,t)
    \ge
    \widetilde u(\hat\rpol(\lvertx\rvert),t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sub_sub_dens_nn}
    \supp v(t)
    \supset
    B_{\tilde R(t)}(0)
    \,,
    \quad
    \log\tilde R(t)
    \sim
    \log t
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  The constants involved in \eqref{eq:sub_sub_dens_n}--\eqref{eq:sub_sub_dens_nn} depend on the parameters appearing in the assumptions and on $v(x,0)$.
\end{theorem}

\begin{remark}
  \label{r:rad_N}
  Clearly the sup estimate valid for $v$ as in Theorems
  \ref{t:sup_sup_dens} and \ref{t:sub_sub_dens} is the same as in
  \eqref{eq:sup_decay}, which should be compared with the result valid for $\rho(s)\sim s^{-\alpha}$, $\alpha>p$, that is the universal bound $t^{-1/(p+m-3)}$ (see \cite{Andreucci:Tedeev:2021b}). Our present bound however depends on the initial data, as the known bound valid in the subcritical case $\alpha<p$, whose decay rate is $t^{-(N-\alpha)/K}$, $K=(N-\alpha)(p+m-3)+p-\alpha$.
\end{remark}

\textbf{Plan of the paper.}
Section~\ref{s:pre} contains some elementary auxiliary results.
Then Proposition~\ref{p:main_sup_sub} is proved in Sections \ref{s:sss} (supersolutions) and \ref{s:ssb} (subsolutions).
Finally in Section~\ref{s:rad} we carry out the transformation of variables linking \eqref{eq:pde} and \eqref{eq:rad_N}.

The post-theorem text gives the following proof sketch for Theorems~\ref{t:sup_sup} and \ref{t:sub_sub}: “The proof follows straightforwardly from an application of the method given in the proof of \cite[Theorem~1]{Tsutsumi:1988}, when we take into account Proposition~\ref{p:main_sup_sub} and Lemmas \ref{l:sss_comp} and \ref{l:ssb_comp}.”

The post-theorem text gives the following proof sketch for the main theorem and
\begin{theorem}
  \label{t:sub_sub}
  Assume \eqref{eq:degen} and \eqref{eq:powerlike}. Let $u$ be a strong solution to \eqref{eq:pde}--\eqref{eq:initd} with a compactly supported and bounded data $u_{0}$, such that $u_{0}(x)\ge \eps>0$ for $\abs{x}\le\ell$, $\ell>0$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\ups$ as in \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a subsolution and
  \begin{equation}
    \label{eq:sub_sub_n}
    u(x,t)
    \ge
    \ups(x,t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,.
  \end{equation}
  We have also
  \begin{equation}
    \label{eq:sub_sub_nn}
    \supp u(t)
    \supset
    B_{R(t)}(0)
    \,,
    \quad
    R(t)
    =
    \gamma
    \exw^{(-1)}(\log t)
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  for a suitable constant $\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\eps$, $\ell$.
\end{theorem}
: “The proof follows straightforwardly from an application of the method given in the proof of \cite[Theorem~1]{Tsutsumi:1988}, when we take into account
\begin{proposition}
  \label{p:main_sup_sub}
  The constants $C_{*}$, $\varGamma$, $t_{0}> 1$, $\rpol_0$ and
  $\nu_{0}$ can be selected so that $\ups$ as defined in
  \eqref{eq:sss_sup}--\eqref{eq:sss_ups_2} is a strong supersolution to
  \eqref{eq:pde}. Alternatively, we may select such constants so that
  $\ups$ is a strong subsolution to \eqref{eq:pde}.
\end{proposition}
 and
\begin{lemma}
  \label{l:sss_comp}
  Let $L>0$, $M>0$ be given positive numbers. Then it is possible to select $\rpol_{0}$, $C_{*}$, $t_{0}$, $\varGamma$ so that $\ups$ is a supersolution and
  \begin{equation}
    \label{eq:sss_comp_n}
    \ups(x,0)
    \ge
    M
    \,,
    \qquad
    \abs{x}\le L
    \,.
  \end{equation}
\end{lemma}
 and
\begin{lemma}
  \label{l:ssb_comp}
  Let $\ell>0$, $\eps>0$ be given positive numbers. Then it is possible to select $\rpol_{0}$, $C_{*}$, $t_{0}$ and $\varGamma$ so that $\ups$ is a subsolution and
  \begin{gather}
    \label{eq:ssb_comp_n}
    \supp \ups(0)
    \subset
    \{\abs{x}\le \ell\}
    \,,
    \\
    \label{eq:ssb_comp_nn}
    \ups(x,0)
    \le
    \eps
    \,,
    \quad
    x\in\RN
    \,.
  \end{gather}
\end{lemma}
.”

\label{t:sup_sup}
  \begin{equation}
  \label{eq:degen}
  1<p<N
  \,,
  \qquad
  p+m-3>0
  \,,
\end{equation}
  and
  \begin{equation}
  \label{eq:powerlike}
  \alpha_{1}
  \frac{\exw(s)}{s}
  \le
  \exw'(s)
  \le
  \alpha_{2}
  \frac{\exw(s)}{s}
  \,,
  \qquad
  s>0
  \,,
\end{equation}
  hold. Let $u$ be a strong solution to
  \begin{alignat}{2}
  \label{eq:pde}
  \ew(x)
  \pder{u}{t}
  -
  \Div\big(
  \ew(x)
  u^{m-1}
  \abs{\grad u}^{p-2}
  \grad u
  \big)
  &=
    0
    \,,
  &\qquad&
           \text{in $S_{T}$,}
  \\
  \label{eq:initd}
  u(x,0)
  &=
    u_{0}(x)
    \,,
  &\qquad&
           x\in\RN
           \,,
\end{alignat}
  with a compactly supported and bounded data $u_{0}$.
  \\
  Then there exist constants $t_{0}$, $C_{*}$, $\rpol_{0}$ and $\varGamma$ such that $\widetilde u$ defined by
  \begin{alignat}2
  \label{eq:sss_sup}
  \ups(x,t)
  &=
    \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
    \Big[
    \tar(\tau)^{\frac{1}{p-1}}
    -
    \rat(\rpol)^{\frac{1}{p-1}}
    \Big]^{\frac{p-1}{p+m-3}}_{+}
    \,,
    &\quad&
            \rpol\ge \rpol_{0}
            \,,
  \\
  \label{eq:sss_ups_2}
  \ups(x,t)
  &=
    \frac{C_{*}}{(t+t_{0})^{\frac{1}{p+m-3}}}
    \Big[
    \tar(\tau)^{\frac{1}{p-1}}
    -
    \ram(\rpol)
    \Big]^{\frac{p-1}{p+m-3}}_{+}
    \,,
    &\quad&
            \rpol<\rpol_{0}
            \,,
\end{alignat}
  is a supersolution and
  \begin{equation}
    \label{eq:sup_sup_n}
    u(x,t)
    \le
    \widetilde u(x,t)
    \,,
    \qquad
    (x,t)\in S_{\infty}
    \,,
  \end{equation}
  holds.
  We have also
  \begin{equation}
    \label{eq:sup_sup_nn}
    \supp u(t)
    \subset
    B_{R(t)}(0)
    \,,
    \quad
    R(t)
    =
    \gamma
    g^{(-1)}(\log t)
    \,,
    \qquad
    t>t_{0}
    \,,
  \end{equation}
  for a suitable constant $\gamma>0$. The constants here depend on the parameters appearing in the assumptions and on $\sup_{\R^{\SpDim}} u_{0}$, $R(0)\,.

Existential–Universal
Inequality or Bound

Question: Assume 1 < p < N and p + m - 3 > 0. Let g:[0,∞)→[0,∞) satisfy g(0)=0, g(s)>0 for s>0, and, for some 0 < α1 ≤ α2 < p, α1 g(s)/s ≤ g'(s) ≤ α2 g(s)/s for every s>0. Set f(x)=exp(g(|x|)). Let u be a nonnegative strong solution of the Cauchy problem f(x)u_t - Div(f(x)u^{m-1}|∇u|^{p-2}∇u)=0 in S_T=R^N×(0,T), with bounded compactly supported initial datum u0 (say supp u0⊂B_{R(0)}(0)). Also define G(r)=∫_0^r g(s)/s ds, J(r)=r^p/G(r), τ=Γ log(t+t0), E(τ)=J(G^{-1}(τ))=(G^{-1}(τ))^p/τ, and I(r)=ν0 (r^p/G(r0))^{1/(p-1)} + (1-ν0)J(r0)^{1/(p-1)}, where ν0 is the fixed constant used in the construction. Which statement holds?

There exist constants t0, C*, r0, Γ, and a constant γ>0, depending only on the parameters in the assumptions and on sup_{R^N} u0 and R(0), such that the radial function u~(x,t), with r=|x|, defined by u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - J(r)^(1/(p-1))]_+^((p-1)/(p+m-3)) for r≥r0, and u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - I(r)]_+^((p-1)/(p+m-3)) for r<r0, is a supersolution, and u(x,t) ≤ u~(x,t) for all (x,t) in S_∞=R^N×(0,∞). Moreover, for every t>t0, supp u(t) ⊂ B_{R(t)}(0) with R(t)=γ g^{-1}(log t).

Choices:
B. There exist constants t0, C*, r0, Γ, and a constant γ>0, depending only on the parameters in the assumptions and on sup_{R^N} u0 and R(0), such that the radial function u~(x,t), with r=|x|, defined by u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - J(r)^(1/(p-1))]_+^((p-1)/(p+m-3)) for r≥r0, and u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - I(r)]_+^((p-1)/(p+m-3)) for r<r0, is a supersolution, and u(x,t) ≤ u~(x,t) for all (x,t) in S_∞=R^N×(0,∞). Moreover, for every t>0, supp u(t) ⊂ B_{R(t)}(0) with R(t)=γ g^{-1}(log t).
C. There exist constants t0, C*, r0, and Γ such that the radial function u~(x,t), with r=|x|, defined by u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - J(r)^(1/(p-1))]_+^((p-1)/(p+m-3)) for r≥r0, and u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - I(r)]_+^((p-1)/(p+m-3)) for r<r0, is a supersolution, and u(x,t) ≤ u~(x,t) for all (x,t) in S_∞=R^N×(0,∞).
D. There exist constants t0, C*, r0, Γ, and a constant γ>0, depending only on the parameters in the assumptions, such that the radial function u~(x,t), with r=|x|, defined by u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - J(r)^(1/(p-1))]_+^((p-1)/(p+m-3)) for r≥r0, and u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - I(r)]_+^((p-1)/(p+m-3)) for r<r0, is a supersolution, and u(x,t) ≤ u~(x,t) for all (x,t) in S_∞=R^N×(0,∞). Moreover, for every t>t0, supp u(t) ⊂ B_{R(t)}(0) with R(t)=γ g^{-1}(log t).
E. There exist constants t0, C*, r0, Γ, and a constant γ>0, depending only on the parameters in the assumptions and on sup_{R^N} u0 and R(0), such that the radial function u~(x,t), with r=|x|, defined by u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - J(r)^(1/(p-1))]_+^((p-1)/(p+m-3)) for r≥r0, and u~(x,t)=C*(t+t0)^(-1/(p+m-3)) [E(τ)^(1/(p-1)) - I(r)]_+^((p-1)/(p+m-3)) for r<r0, is a subsolution, and u(x,t) ≥ u~(x,t) for all (x,t) in S_∞=R^N×(0,∞). Moreover, for every t>t0, supp u(t) ⊂ B_{R(t)}(0) with R(t)=γ g^{-1}(log t).