How to Solve NonLinear PDEs

Hi everyone!
I'm new in Comsol (I started just 2 days ago). I read that Comsol is powerful because you can add your own PDE in the model.

I'm trying to use the "equation based model" to solve my own NonLinear PDEs. I attached what kind of NL PDE I want to solve.
In order to start, I simplified the problem, and I solved the following case:

1) well-known diffusion equation (equation (2) in attached file)
2) eq (2) with space-dependent forcing term (equation (3) in attached file)
3) eq (2) with space-and-time-dependent forcing term (equation (4) in attached file)

I matched the result with my benchmark using Mathematica. Now I'm not able to do the following:
1) impose Dirichlet BC
2) add the nonlinear term

Moreover I would like to write a system of coupled PDEs. How can I do this?

Finally, the last doubt: I read "weak form" is the most powerful tool to solve nonlinear PDE. However, it is not clear to me how to write the problem in this form. I know the variational principle, but I haven't understood who is the "weak" term in the formulation. The documentation has no examples so it's hard. Have you some well-made tutorial for equatio-based modeling? I was not able to find them over the web.

Thanks in advance for all your answers and suggestions.
Pietro