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:
$(G(t)\ast u_0)(x) = \int_{\mathbb{R}^{N}} G(t,y)u_0(x-y) \ \text{d}y$