Uniqueness of the fundamental solution of a 2nd order linear parabolic PDE

Question

I'm reading Avner Friedman, Partial Differential Equations of Parabolic Type. Let \begin{eqnarray} L:=a_{ij}\dfrac{\partial}{\partial x_i\partial x_j}+b_i\frac{\partial}{\partial x_i}+c-\frac{\partial}{\partial t} \end{eqnarray} be a uniformly parabolic differential operator. Is the fundamental solution of equation $Lu=0$ unique under the following condition?

Let $D$ be a domain in $\mathbb{R}^n$, $T_0<T_1, 0<\alpha<1$. $\Omega:=\bar{D}\times[T_0,T_1]$. $a_{ij}, b_i, c:\Omega\to\mathbb{R}$ are bounded and $\exists A>0$ s.t. $\forall x,y\in D, t,s\in[T_0,T_1]$ \begin{eqnarray} |a_{ij}(y,s)-a_{ij}(x,t)|\leq A\left(|y-x|+\sqrt{|s-t|}\right)^\alpha \\ |b_i(y,t)-b_i(x,t)|\leq A|y-x|^\alpha \\ |c(y,t)-c(x,t)|\leq A|y-x|^\alpha \end{eqnarray}

Answer 1

If $LF=\delta$ and $Lu=0\,,$ then $L(F+u)=\delta\,,$ and it would seem that $L$ has nonzero kernel. Are there conditions on $F$?