Uniqueness of solution on Advection Diffusion equation

Question

Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following Advection-diffusion Equation given by \begin{align} \nabla\cdot\left(\mathbf{V}\psi-D\nabla \psi\right)&=F\quad \text{in}\quad \Omega\times(0,\,T)\\ \psi(x,0)&=0\quad \text{in}\quad \Omega\\ \psi(0,t)&=0\quad \text{in}\quad \Omega\\ \nabla\psi(\ell,t)&=0\quad \text{on}\quad \partial \Omega\times(0,\,T) \end{align} I would like to show the uniqueness of the solutions. I assume existence of two different solutions $\psi_{1}$ and $\psi_{2}$ and defined the difference of the solutions $\omega=\psi_{2}-\psi_{1}$ clearly $\omega$ satisfies the equation \begin{align} \nabla\cdot\left(\mathbf{V}\omega-D\nabla \omega\right)&=0\quad \text{in}\quad \Omega\times(0,\,T)\\ \omega(x,0)&=0\quad \text{in}\quad \Omega\\ \omega(0,t)&=0\quad \text{in}\quad \Omega\\ \nabla\omega(\ell,t)&=0\quad \text{on}\quad \partial \Omega\times(0,\,T) \end{align} I considered the following integral \begin{align} J=\int\limits_{\Omega}\omega\nabla\cdot\left(\mathbf{V}\omega-D\nabla \omega\right) dV=0 \end{align} My question is I want to show that $J>0$ so that I conclude that if $J>0$ then all the terms inside the inside the integration are zero. Thus $\psi_{1}=\psi_{2}$. I do not know if this is possible or how to do this.