The $L^p$ Norm of the Heat Kernel and its Divergence and Gradient I am working with the Heat Kernel defined on Euclidean space with dimension $N$ as:
$G(t,x) := \frac{1}{(4 \pi t)^{-N/2}} e^{-|x|^{2}/4t}$
I have been told by my professor, and found on this post the following expression for the $L^p$ norm of the Heat Kernel:
$||G(t)||_{L^{p}(\mathbb{R}^{N})} = C_{p} t^{(-N/2)(1-\frac{1}{p})}$, where $C_p$ is some positive constant, $1 < p < \infty$.
I also have an expression for the norm of the gradient:
$||\nabla G(t)||_{L^{p}(\mathbb{R}^{N})} = C_p t^{-1/2}$.
My first question is simply where I can find a source with a proof of these expressions! I have been unable to find them on Google...
My second question is whether or not we have an upper bound on the norm of the divergence of $G$. Specifically, I need an estimate for the following expression:
$|| a \cdot \nabla G(t)||_{L^{1}(\mathbb{R}^{N})}$, where $a \in \mathbb{R}^{N}$ constant. I presume we could easily deal with this constant by considering: 
$^{\text{max}}_{1 \leq i \leq N} \ a_{i} ||\text{div }G(t)||_{L^{1}(\mathbb{R}^{N})}$. So we just need an estimate for the $L^1$ norm of $\text{div }G(t)$.
The end goal of the problem I am working on is to show that the following map $\phi$ is a contraction map on $C([0,T]; L^{1}(\mathbb{R}^{N}) \cap L^{\infty}(\mathbb{R}^{N}))$ equipped with the norm $||u|| = ^{\text{sup}}_{0<t<T}(||u||_{L^{1}(\mathbb{R}^{N})} + ||u||_{L^{\infty}(\mathbb{R}^{N})})$.
$\phi(u)(t) := G(t) \ast u_0 + \int^{t}_{0} a \cdot \nabla G(t-s) \ast (|u(s)|^{q-1}u(s)) \text{d}s $. Here, $u_0 \in L^{1}(\mathbb{R}^{N}) \cap L^{\infty}(\mathbb{R}^{N})$ is the time-initial value of $u$, and $\ast$ denotes convolution of functions:
$(G(t)\ast u_0)(x) = \int_{\mathbb{R}^{N}} G(t,y)u_0(x-y) \ \text{d}y$
 A: The first of these is simply a rescaling:
$$ \lVert G(t,\cdot) \rVert_p^p = \int_{\mathbb{R}^N} (G(t,x))^p \, dx = \frac{1}{(4\pi t)^{Np/2}}\int_{\mathbb{R}^N} e^{-\lvert x \rvert^2 p/4t} \, dx = \frac{1}{(4\pi t)^{Np/2}} (t/p)^{N/2} \int_{\mathbb{R}^N} e^{-\lvert y \rvert^2 /4} \, dy $$
with $x = \sqrt{t/p} y $. The latter integral is just $ (4\pi)^{N/2} $ by evaluating in each coordinate separately, so we get
$$ \lVert G(t,\cdot) \rVert_p^p = (4\pi)^{N/2-Np/2}  t^{N/2-Np/2} \\
\lVert G(t,\cdot) \rVert_p = (4\pi t)^{-(N/2)(1-1/p)} . $$
The precise constant in $\lVert G(t,\cdot) \rVert_p$ depends on the norm used for $\mathbb{R}^N$, but we still have
$$ \nabla G(t,x) = \frac{x}{2t} G(t,x) . $$
Thus
$$ \lVert \nabla G(t,\cdot) \rVert_p^p = \int_{\mathbb{R}^N} \left\lVert \frac{x}{2t} G(t,x) \right\rVert^p dx = \int_{\mathbb{R}^N} \lVert x \rVert^p \frac{1}{(2t)^p} G(t,x)^p \, dx , $$
since $\lVert \cdot \rVert$ is a norm, and using the same change of variables as before gives
$$ \lVert \nabla G(t,\cdot) \rVert_p^p = 2^{-p} t^{-p} (4\pi t)^{-Np/2} (t/p)^{p/2} (t/p)^{N/2} \int_{\mathbb{R}^N} \lVert y \rVert^p e^{-\lvert y \rvert^2/4} \, dy , $$
where the power of $t$ is actually $-p-Np/2+p/2+Np/2 = -p(N/2)(1-1/p)-p/2$. Hence
$$ \lVert \nabla G(t,\cdot) \rVert_p \propto t^{-(N/2)(1-1/p)+1/2} . $$
(For the Euclidean norm $\lvert x \rvert$, the integral can be done using polar coordinates.)
The easiest way to bound $ \lVert a \cdot \nabla G(t,\cdot) \rVert  $ is probably just to use Cauchy–Schwarz: $ \lvert a \cdot \nabla G(t,x) \rvert \leq \lvert a \rvert \lvert \nabla G(t,x) \rvert $, and then the bound follows from a special case of the above.
