Let's consider $\Omega \subset \mathbb R^3$ , $T\ge 0 , \Omega_T=\Omega \times(0,T]$
Consider the problem
$ \left\{ \begin{align} &u_t-\Delta u+u^3=f,\qquad&&\text{on}\;\Omega_T,\\ &u=0,\qquad&&\text{for}\;x\in\partial \Omega,\;t\ge0,\\ &u=g,\qquad&&\text{for}\;x\in \Omega,\;t=0, \end{align} \right. $
where $f$ and $g$ belong to $L^2(\Omega_T)$ and $L^2 (\Omega)$ respectively.
How can I show that there exists weak solution using Galerkin's approximation?
Any kind of help will be appreciated. Thank you.