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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.

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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.

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    $\begingroup$ There are numerous books covering local existence theory for boundary value problems for second order quasi-linear (and hence, semi-linear) parabolic partial differential equations, as described above. The classics by Friedman "Partial differential equations of parabolic type" and Ladyzenskaya, Ural'ceva, Solonnikov "Linear and Quasilinear equations of parabolic type" contain relavant theory. A More recent book by Lieberman "Second Order parabolic differential equations" is also pretty good, amongst others. $\endgroup$ – JCM Mar 30 '16 at 13:37

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