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I'm considering the following type of PDE: $u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$

with periodic boundary conditions $u_{x}(0)=u_{x}(1)=0$.

Does anyone know of literature/articles I can look at to possibly show local existence(or blow up?). I've spent quite a bit of time looking through literature in particular Henry's Geometric theory of Semi-Linear Parabolic equations and papers by H.Amman on Quasi linear Parabolic equations. Also due to the singularity I considered a weighted sobolev space approach but had no success. I apologize if I'm too vague.

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I would try to find the Lie Symmetry groups for this equation first. A good book on this is Bluman, Cheviakov, Anco. The easiest to find them is to use some computer algebra packages such as Maple and e.g. the related GeM package from Cheviakov.

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