Hello.

I want to prove the existence of a weak solution to:

Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that $$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$ with $u(x, 0) = u_0(x)$ where $u_0 \in C^{1+\alpha}(S^1)$ (and is non-negative) and $n_1$ and $n_2$ are fixed integers (eg. $n_1 = 2$, $n_2 = 3$, which I fix for now).

Also, $u$ should lie in the space $C^{2+\alpha, 1+\alpha}(S^1 \times [0, T-\epsilon])$ for $\epsilon > 0$.

How to prove short-time existence for this PDE? Every book (Michael Taylor's Nonlinear.., Ladyzhenskaya) I look at has existence results that require strong parabolicity (i.e., require $u^2\eta^2 \geq C\eta^2$ for all non-zero $\eta$, which I cannot say I have).

A reference to where this kind of PDE is proven would be appreciated or a detail of how to prove such PDEs.

PS: This question is on Math.SE for a while but it has not received much attention, so I hope it is OK to post it here as otherwise I would be very stuck. Thanks.