We consider the evolution PDE $$ \partial_t f=\operatorname{div}(A \nabla f) $$ on the unknown $f=f(t, x), t \geq 0, x \in \mathbb{R}^d$, with $A=A(x)$ a symmetric, uniformly bounded and coercive matrix, in the sense that $$ \nu|\xi|^2 \leq \xi \cdot A(x) \xi \leq C|\xi|^2, \quad \forall x, \xi \in \mathbb{R}^d . $$
It is worth emphasizing that we do not make any regularity assumption on $A$. We complement the equation with an initial condition $$ f(0, x)=f_0(x) . $$
- Existence. What strategy can be used in order to exhibit a semigroup $S(t)$ in $L^p\left(\mathbb{R}^d\right)$, $p=2, p=1$, which provides solutions to $(0.1)$ for initial date in $L^p\left(\mathbb{R}^d\right)$ ? Is the semigroup positive? mass conservative? So for the existence of semi group , i try to regularize A (but i dont know how to do it , i thought about mollifier but since i have no assuption about A i cant do that)and if i succed i use viscocity method i mean by that considering the equation by approximation and then make $\epsilon$ goes to$\infty$ if any one has hint ? thanks