In this paper Pinsky shows existence, uniqueness and regularity for the problem $$ u_t=\Delta u-a(x) u^p |\nabla u|^q $$ where $a\in C^2( \mathbb{R}^d)$ satisfies the condition $ a(x)|\leq (1+|x|^2)^N$.
His idea is to consider an approximation of the problem in the form
$$
u_t=\Delta u-a(x)(1+\delta|x|^2)^{-N} (u+\varepsilon)^p |\nabla u| ^q
$$ and then suppress the dependence on $\delta$ and $\varepsilon$.
I'm interested in getting a good understanding of the regularity argument. He defined
$$
\beta (x) = a(x)(1+\delta|x|^2)^{-N},
$$ which by hypothesis is a bounded function, and due to (2.7) in the reference linked, it follows that $u$ and $|\nabla u|$ are bounded. Then using the definition of $\nabla u$ as a mild solution, he proves the estimate
$$
\begin{split}
|\nabla u(x+z,t) - \nabla (x,t)| & \leq \int_{\mathbb{R}^d}p(t,x,y)|\nabla \phi(y+z) -\nabla \phi (y)|dy \\ &\quad+ c \int_{0}^t \int_{\mathbb{R}^d}| \nabla_x p(s,x+z,y) - \nabla_x p(s,x,y)|dyds
\end{split}
$$
where $p(t,x,y)$ is the Heat Kernel in $\mathbb{R}^d$. He then estimates this last integral by
$$ \int_{\mathbb{R}^d}| \nabla_x p(s,x+z,y) - \nabla_x p(s,x,y)|dy \leq Kt^{-(1+\alpha)/2}|z|^{\alpha} \mbox { with }0<\alpha<1. $$
He cites Ben Artzi's article for this last estimate. In fact in his proposition 2.4, equation (2.11) he directly gives the above inequality saying only that he is interpolating in (2.6). I don't know what this interpolation argument he used means, I don't know if it is the Mean Value Theorem that he is using in the gradient, as he got this $0<\alpha<1$ moreover I don't know how. I would be grateful if someone could explain me! After that he concludes the argument using a standard parabolic regularity result.
Another question I was thinking about is the following. If this interpolation step is proved by using the Mean Value Theorem, essentially the limitation is $$ \begin{split} \int_{\mathbb{R}^d}| \nabla_x p(s,x+z,y) & - \nabla_x p(s,x,y)|dy \\ &\approx \int_{\mathbb{R}^d} |\nabla_x^2 p(s,x+z ,y)||z| dy \\ &\approx |z| \int_{\mathbb{R}^d} |\nabla_x^2 p(s,x+z,y)| dy. \end{split} $$ It is known that the Hessian of the heat kernel is normally limited by the heat kernel itself, that is $$ |\nabla_x^2 p(s,x+z,y)| \leq Cs^{-1} p(2s,x+z,y) $$ (according to this MathOverflow Q&A). So my second doubt is whether this regularity step could be done differently in more general cases, Imagine that $a(x)=|x|^{-N}$ and if it is possible to find the existence of a solution such that $u $ and $|\nabla u|$ are limited. The change would occur here in the part where the kernel gradients are estimated $$ \begin{split} \int_{\mathbb{R}^d}| \nabla_x p(s,x+z,y) & - \nabla_x p(s,x,y)||y|^{-N}dy \\ & \approx |z| \int_{\mathbb{R}^d} |\nabla_x^2 p(s,x+z,y)||y|^{-N} dy \\ & \approx |z|Cs^{-1} \int_{ \mathbb{R}^d} |p(2s,x+z,y)||y|^{-N} dy. \end{split} $$ But the last integral can be convergent if $N<d$. I have doubts whether this thought is correct, as I have seen papers that say if the nonlinearity has a singularity, the solution cannot be classical. I appreciate any suggestions and comments.
