Lipschitzness of product map on $L^2 \times L^2$

Question

Is the map $(u,v) \rightarrow uv$ a Lipschitz map from $L^2(\Omega) \times L^2(\Omega) \rightarrow L^2(\Omega)$ for $\Omega \subset \mathbb{R}^n$?

I am particularly interested in existence of local solutions of PDEs of the form by arguing that the nonlinearity in the following PDE is locally Lipschitz.

$u_t = \Delta u -a uv $

$v_t = \Delta v +a uv $