I am following up on the answer of Denis Serre to this same question here [Short time existence on nonlinear parabolic PDE](https://mathoverflow.net/questions/99994/short-time-existence-on-nonlinear-parabolic-pde)


I have tried to generalise the proof of the Picard-Lindelof theorem, as suggested by Denis, to prove short time existence as follows:


For the system
$$ (\partial\_t -L)u\_t=f(u,\nabla u) \hskip20pt \text{on} \hskip5pt \Omega \times [0,T] $$
$$ u=u\_0 \hskip20pt \text{on} \hskip5pt \Omega\times \{0\} $$
We can reformulate as $u\_t = e^{-tL}u\_0+\int\_0^te^{-(t-s)L}f(u\_s,\nabla u\_s)ds$


Defining a map $ \Gamma \colon \mathcal{C}([0,T];X)\to\mathcal{C}([0,T];X)$, where $X=(\mathcal{C}(\Omega),||\cdot||\_\infty)$, by 
$$\Gamma(u)\_t= e^{-tL}u\_0+\int\_0^te^{-(t-s)L}f(u\_s,\nabla u\_s)ds $$
Then I am able to show $\Gamma$ is well defined and a contraction mapping for $T$ sufficiently small, to deduce short time existence, under the following conditions on $f\colon X \to X$


(I) For all $u\in \mathcal{C}([0,T];X)$, $\int\_0^T||f(u\_s,\nabla u\_s)||\_\infty ds<\infty$, so $\Gamma$ is well defined. 


(II) f is Lipshitz on $\Omega$ with respect to its first argument, $||f(u,\nabla u)-f(v,\nabla v)||\_\infty \leq K||u-v||\_\infty$, so $\Gamma$ is a contraction mapping for $0<T<K^{-1}$


I haven't been able to find an argument for loosening these conditions on $f$, which is not good enough as I would like to allow $f$ to take the form of a quadratic in the gradient as in harmonic map flow. Would greatly appreciate any advice on the problem!


Many Thanks,
A.



Proving short time existence for semi-linear parabolic PDE
You would recommend the beautiful book of M. TaylorPartial Differential Equations, Vol 3 : Nonlinear Equations, Applied Math Sciences series, Vol 117, Springer. In particular the chapter 15 :  Nonlinear Parabolic Equations.


