$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal regularity of solutions to the Neumann heat equation $$ \begin{cases} \partial_t u=\Delta u & \text{in }\Omega\\ \partial_\nu u=0 & \text{on }\partial\Omega\\ u(0,x)=u_0(x) \end{cases} $$ for some nonnegative initial datum $u_0\in L^1_+(\Omega)$. If that helps I actually know in addition that $u_0\in L^1\log L^1(\Omega)$, and in fact that $u_0$ has finite Fisher information $\int_\Omega u_0|\nabla\log u_0|^2=4\int_{\Omega}|\nabla\sqrt {u_0}|^2<+\infty$, but I do not want to (cannot!) assume any standard Sobolev regularity.
Several questions are of interest to me:
- What is the optimal regularity of $u$, both in time and space?
- Is this regularity enough to guarantee uniqueness in any sense? If not, can one somehow select a good notion of minimally regular solution to recover uniqueness?
- Is this unique solution representable via Green's functions/integral representation, $u(t,x)=\int_\Omega K_t(x,y) u_0(y) dy$? (some probabilistic construction, perhaps? in which case the regularity from question 1 may boil down to potential theory?)
- Is it true that $u(t,x)\geq c_t>0$ uniformly in $x$ as soon as $t>0$?
I have digged a little bit but this seems to be suprisingly not so well documented, and I could not conclude anything after spending one full day of bibliographical research (at least for the Neumann problem, the Dirichlet version of the question returns many more answers). Any help, hints, or references would be highly appreciated!
