I am following up on the answer of Denis Serre to this same question here 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.
