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I am curious what kinds of techniques one uses to show existence of PDEs with nonlinearities.

I am aware of: 1) Minimisation problems

2) Semigroup

(both of which I'd like to avoid)

For linear PDEs, Galerkin method is often used. Does this work for nonlinear too? Can someone point me to literature that addresses this for parabolic PDEs? Thanks

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Did you check Evans Chapter 9 section 1&2 about nonvariational techniques? –  Shuhao Cao Aug 19 '13 at 2:16
    
@ShuhaoCao Yeah I did. I like Galerkin method though. –  michael_faber Aug 23 '13 at 20:15

1 Answer 1

The Galerkin method can definitely be used for nonlinear problems.

I have seen often that one can couple Galerkin method with a fixed point method. For example you have a nonlinear PDE which you can linearise. The linearised PDE you can solve maybe with a Galerkin method, and then one shows that there is a fixed point of the appropriate map that takes the part you made linear to the solution.

Here is a paper where a Galerkin method is applied to a porous medium equation.

Also people use a famous theorem by Crandall and Liggett for nonlinear PDEs where basically the PDE is discretised in time and one gets a series of elliptic problems. One then shows that the solutions to these problems converge in some sense to a mild solution if I recall correctly. But I think this has something to do with semigroups.

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