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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 $u_{x}(0,t)=u_{x}(1,t)=0$, $u(x,0)=u_{0}:S^{1}\rightarrow\mathbb{R}.$

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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  • $\begingroup$ It may be worth having a look in the book "Second order parabolic differential equations" by G. Lieberman, however, it may not. Good luck. $\endgroup$
    – JCM
    Commented Sep 9, 2013 at 21:06

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