Consider the Cauchy problem for the nonlinear diffusion equation (or the porous medium equation): \begin{equation}\nonumber%\label{eqn: LV 2-species scaled convolution type} \textbf{(P)} \begin{cases} %\vspace{3mm} \dfrac{\partial u}{\partial t}=\dfrac{\partial}{\partial x}\left(u^{\beta}\,\dfrac{\partial u}{\partial x}\right),\ \ &(x,t)\in\mathbb{R}\times(0,\infty),\\ %\vspace{3mm} u(x,0)=f(x)\ge0,\ \ &x\in\mathbb{R},%\\ \end{cases} \end{equation} where $u=u(x,t)$ and $\beta>0$ is a constant.
(1) It is well-known that when $\beta=0$, $\textbf{(P)}$ becomes the Cauchy problem for the heat equation, and it allows an representation integral formula: \begin{equation} u(x,t)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{4\,\pi\,t}}e^{-\frac{(x-y)^2}{4\,t}}\,f(y)\,dy. \end{equation} See for instance, http://www.math.ualberta.ca/~xinweiyu/527.1.08f/lec12.pdf. However, when $\beta\neq0$, does $\textbf{(P)}$ admit an representation formula?
(2) Suppose that $u=u(x,t)$ solves $\textbf{(P)}$. Can we show that the following integral
\begin{equation}
\int_{-\infty}^{\infty}
\bigg|
u^{\beta+1}\,\left(\frac{u_{xx}}{u}\right)^5 \left(\frac{u_{xxx}}{u}\right)^2\left(\frac{u_{x}}{u}\right)^3
\bigg|
\,dx<\infty?
\end{equation}
Any reference, suggestion, idea, or comment is welcome. Thank you!