# Arguing for uniqueness and existence in $[0,T]$

Lets consider $\Omega \subset \mathbb R^3$ , $T\ge 0 , \Omega_T=\Omega \times(0,T]$

Consider the problem $u_t-\Delta u+u^3=f$ on $\Omega_T$ $u=0$ for $x\in\partial \Omega , t\ge0$

$u=g$ for $x\in \Omega , t=0$ where $f$ and $g$ belong to $L^2(\Omega_T)$ and $L^2 (\Omega)$ respectively. Consider $u_m = \sum _{k=1}^m d_k(t)w_k$ be the galerkin approximation. How do i argue for uniqueness and existence of solution in $[0,T]$ ? Thank you for your help.

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