Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $.  Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, assume that $u_t$ and $u'_t$ satisfies the generalized porous medium equations 
\begin{align*}
\partial_t u_t= \Delta (F\circ u_t +\varepsilon u_t),& \qquad u_0= v\\
\partial_t u'_t= \Delta (F\circ u'_t),&  \qquad u_0= v
\end{align*}
in the weak sense. 

1. Does it holds that in the Fourier variable we have the following comparison $|\widehat{u'_t}(\xi)|\leq|\widehat{u_t}(\xi)| $ for all $\xi\in \Bbb R^d$ (or the converse inequality)?<br>  By passing into Fourier variables and differentiating $|\widehat{u_t}(\xi)|^2 $  and $|\widehat{u'_t}(\xi)|^2$ respectively, gives  
\begin{align*}
\partial_t |\widehat{u_t}(\xi)|^2
&= -2|\xi|^2\operatorname{Re}\big(\overline{\widehat{u_t}}(\xi) \widehat{F\circ u_t}(\xi)\big)
-2\varepsilon|\xi|^2 |\widehat{u_t}(\xi)|^2\\
\partial_t |\widehat{u'_t}(\xi)|^2&= -2|\xi|^2\operatorname{Re}\big(\overline{\widehat{u'_t}}(\xi) \widehat{F\circ u'_t}(\xi)\big)
\end{align*}
with $\widehat{u_0}(\xi)= \widehat{u'_0}(\xi)= \widehat{v}(\xi)$. Therefore it would be enough to show compare $\partial_t |\widehat{u'_t}(\xi)|^2$ and $\partial_t |\widehat{u_t}(\xi)|^2. $
\begin{align*}
\partial_t |\widehat{u'_t}(\xi)|^2
\leq \partial_t |\widehat{u_t}(\xi)|^2. 
\end{align*} 

2. For $u\geq0$ and integrable, do we have that 
\begin{align*}
\operatorname{Re}\big(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi)\big)\geq0\qquad\text{for all $\xi\in \Bbb R^d$ ?}
\end{align*}
Note that for $\tilde{u}(x)=u(-x)$,
\begin{align*}
\operatorname{Re}\big(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi)\big)= \operatorname{Re}\big( \widehat{\tilde{u}* F\circ u}(\xi)\big)= \widehat{\tilde{u}* F\circ u}(\xi)\big)+ \widehat{F\circ  \tilde{u}* u}(\xi)\big). 
\end{align*}

Any idea, partial answer or reference it warmly appreciated.