Use the following definition for lower semi continuity. That is, $f$ is lower semi continuous at $x_0$ is $\liminf_{x\rightarrow x_0} f(x) \geq f(x_0)$. This is equivalent to

For all $\epsilon > 0$ there exists $\delta > 0$ so that $\epsilon \geq f(x_0) - f(x)$ for all $x \in B_\delta(x_0)$

Fix $u \in L^2(0,T;L^2(\Omega))$ and pick $\epsilon > 0$. One has \begin{eqnarray} F(u) - F(\frac{1}{2}v) &=& \int_0^T \int_\Omega \left[ f(u) - f(\frac{1}{2}v) \right] \\ &\leq& \int_0^T \int_\Omega f(\frac{1}{2}(2u-v)) \end{eqnarray} Use the fact that convex functions are Lipshitz to get that $$F(u) - F(v) \leq C \int_0^T \int_\Omega \left| u - \frac{1}{2}v\right|$$ for some constant $C$. From this you can easily conclude what you want by using the fact that you are playing over a bounded domain and using Holder.