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.



Existence of solution to quasilinear parabolic PDEs
Use the references on strongly parabolic PDE's to show that for each $\epsilon > 0$, you can solve
$$
\partial\_t u\_\epsilon = (\epsilon + |u\_\epsilon|^{n\_1})\partial\_x^2u\_\epsilon + |u\_\epsilon|^{n\_2}.
$$
Using energy estimates, get estimates for the time of existence and the $L^2$ Sobolev norms of $u$ that are independent of $\epsilon$. Let $\epsilon \rightarrow 0$.


By the maximum principle, $u$ will be strictly positive for small positive time. Therefore, the equation is strictly parabolic for small positive time. This will imply that $u$ is smooth for small positive time.


