Let $T=1$ and $\Omega = (0, \pi) \times (o,\pi)$. We consider the following parabolic PDE:
$$\frac{\partial u}{\partial t}-\Delta u+u_x-u=t\sin(x+y), \text{ on } (0,T) \times \Omega$$
$$\frac{\partial u}{\partial n}+u=1, \text{ on } (0,T)\times \partial \Omega$$
$$u(0,x)=1, \text{ for } x\in\Omega.$$
How can we prove the existence of the solution of such a problem? I tried to solve this using Lax-Milgram Lemma, but in this case, this result can not be applied because the bilinear form from the variational formulation is not symmetric.