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I want to solve the coupled system $$u_t = f(u,u_x,u_{xx})$$ $$v_t = g(u,u_x,u_{xx}, v, v_x, v_{xx})$$ with $f$ and $g$ such that the right hand sides are linear with respect to their arguments.

I am aware that the Ladyzenskaja book covers this, but the notation and typesetting is horrible and I don't want to use it. Can anyone recommend me a more modern book that treats this kind of PDEs?

I just want an existence/uniqueness result in Hölder spaces.

Thanks a lot.

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Did you see Henry's Geometric theory of semilinear parabolic equations? Added: there is also Abstract Parabolic Evolution Equations and their Applications, but I am not very well aware of the content of this one. –  Artem Jul 26 '12 at 16:41
@Artem No I didn't, I'll check them out. Thanks! –  Pink Panther Jul 27 '12 at 7:49

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