Uniqueness of weak solutions to a nonlinear parabolic PDE.

Question

Consider the following nonlinear parabolic PDE:
$$ \begin{cases} u'-div(A(x,u)\nabla u)=f(u) \;\; in \; \Omega, \\ A(x,u)\nabla u\;.\; n=0\;\; on \;\partial \Omega \times]0,T[,\\ u(.,0)=g \; in \; \Omega \times ]0,T[. \end{cases} $$ With the folowing assumptions:
$\Omega \subset \mathbb{R}^n $ is an open bounded set.
$n$ denotes the normal to the boundary $\partial \Omega.$
$f: \mathbb{R}\longrightarrow \mathbb{R}$ is a globally lipschitz continuous function.
$g \in H^1(\Omega).$
$ A(x,t)=a_{ij}(x,t)$ is a matrix satisfying:
$ \ast \;a_{ij}$ is a Carathéodory function.
$ \ast \; a_{ij} \in L^\infty(\Omega \times \mathbb{R})$
$\ast \;\exists \theta>0 \;\;\; \forall \eta,\xi \in \mathbb{R}^n \;\;\; \langle A(x,t) \eta, \xi \rangle \geq \theta \langle\eta,\xi\rangle\;\;\; \forall t \in \mathbb{R}\;\;$ and a.e $\forall x \in \Omega.$

I have been able to prove the existence of weak solutions to this problem, I'm trying to prove the uniqueness of weak solutions by adding more assumptions on the matrix $A$.
Let $u_1$ and $u_2$ be two weak solutions, then $\forall v \in H^1(\Omega)$:
$$\displaystyle \int_{\Omega} (u'_1-u'_2) v\; dx+\int_{\Omega} (A(x,u_1)-A(x,u_2)) (\nabla u_1-\nabla u_2) . \nabla v\; dx=\int_{\Omega} (f(u_1)-f(u_2)) v \;dx.$$ Letting $v=u_1-u_2$ and by using the assumptions on A and f , we get: $$\displaystyle \dfrac{1}{2} \dfrac{d}{dt} \int_{\Omega} \vert u_1-u_2 \vert^2 \; dx+(\theta-\Vert A \Vert_{L^{\infty}}) \int_{\Omega} \vert \nabla u_1-\nabla u_2) \vert^2 \; dx\leq L \int_{\Omega} \vert u_1-u_2\vert^2 \;dx.$$ Now if $(\theta-\Vert A \Vert_{L^{\infty}}) \geq 0$, the uniqueness would've followed easily, but this assumption is impossible to make since $(\theta-\Vert A \Vert_{L^{\infty}})\leq 0$ by definition. Is there an assumption we can make on $A$ to get the uniqueness of solutions?