Fadeo Galerkin method to prouve existence of parabolic problem

Question

we have the following problem $$ \dfrac{\partial u}{\partial t} - \Delta u + F(u)= f(x,t); t >0, x \in \mathbb{R}^n; \ u(x,0)=0 $$ where $F$ is n linear, lipchitzian increazing function.

My question is how we prouve the existence of solution of this problem using Fadeo Galerkin method? I learn books but i don't found application of Fadeo Galerkin in the general case when F is no linear.

Thank's in advance to the help

Answer 1

You can find it in PDEs book of L. C. Evans.

Answer 2

In this case, you can not apply the Faedo-Galerkin method, it is not applicable in a free domain $\mathbb{R}^n$, the spectrum of the Laplacian is not discrete, in your case, there is no eigenvalues and eigenvectors of the Laplacian you can not make the approximation.