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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.

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Galerkin will work, but semigroup approach is easier. – timur Aug 23 '12 at 20:43

What have you use so far? Why Galerkin method?

As for the stationary problem, $-\Delta u + u^3 = f$, it can be reduced to an abstract Hammerstein equation. This is described in a way more general setting in Michael Holst, Ari Stern: Semilinear mixed problems on Hilbert complexes and their numerical approximation, but they say you need a maximum principle for the equation in order to apply their theory.

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