I'm looking for suggestions to solve: $$ u_t = u_{xxxx}-3u(u_x)^2-\frac{3}{2}u^2u_{xx}+\frac{1}{2}u_{xx}+F $$ where $F$ is currently an unknown function of unknown type (might be linear, exponential, etc.). Ignoring that ambiguity (pretend it vanishes), I am unsure of any method to solve such a PDE. I attempted separation of variables but that didn't help because of the nonlinearities. I then tried seeking a steady-state solution however that didn't lead to anything fruitful. After that, I tried using the $1$D and $2$D Fourier Transforms in space and space and time respectively but again that didn't really help. In this case, I got terms that involved self-convolutions of either $u$ or $u_x$ and this made the equation even harder to solve.
One last thought I had was to maybe use the Cole-Hopf Transform: i.e. introduce some new variable $w=\phi(u)$. In doing this, I started calculating partials of $w$ that would appear in my PDE and the expressions I got seemed even more complicated. This kind of leads me to believe that this won't be a fruitful avenue either, however, I have never really tried using this method so I am wondering if I'm maybe doing something wrong. Here I found: $$ w_t = \phi'(u)u_t $$ $$ w_x = \phi'(u)u_x $$ $$ w_{xx} = \phi''(u)u_x^2+\phi'(u)u_{xx} $$ $$ w_{xxx} = \phi'''(u)u_x^3 + 3\phi''(u)u_xu_{xx}+\phi'(u)u_{xxx} $$ $$ w_{xxxx} = \phi^{(4)}(u)u_x^4 + 6\phi'''(u)u_x^2u_{xx}+3\phi''(u)u_{xx}^2+4\phi''(u)u_xu_{xxx}+\phi'(u)u_{xxxx} $$ Now I just don't really see how to use these equations to generate a simplified version of my original PDE or even how to definitively say that this won't help me in reducing the PDE. Any help or advice would be appreciated.
Are you sure that an analytical solution exists? Only very special nonlinear PDEs have analytical solutions. As a first step you could try the Painlevé Integrability Test. This gives necessary but not sufficient conditions for a PDE to be integrable. The test is cumbersome to do by hand but is manageable with computer algebra systems. I found some hits here https://www.swmath.org/?term=Painlev%C3%A9%20integrability that may be useful for you.
If your equation passes the Painlevé test, you'll need to look for a Lax pair. Note that one may not exist since the Painlevé property is necessary but not sufficient for integrability. See this Mathoverflow post for some suggestions.
You'll want to read up on Integrable Systems if you're not familiar with the topic. It's been nearly 15 years since I've studied this area, but I recall Introduction to Classical Integrable Systems [1] by Babelon et al. being a nice read although I recall almost none of its contents.
[1] Babelon, Olivier; Bernard, Denis; Talon, Michel, Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 0-521-82267-X/pbk). xii, 602 p.(2003). ZBL1045.37033.