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I am trying to assert short time existence for a fully nonlinear equation of the general form

\begin{equation} \begin{cases} u_t = F(x,u,Du,D^2u) & \text{in }(0,T)\times\Omega\\ u(\cdot,0) = u_0(\cdot) & \text{in }\Omega \\ u(x,t) = 1 & \text{on }(0,T)\times\partial\Omega \end{cases} \end{equation}

where $u_0$ is smooth, $F=F(x,z,p,A)$ is smooth in all entries and concave in $A$, and $\Omega\subset\mathbb{R}^n$ is a smooth bounded domain. I know that the initial data is uniformly elliptic, in the sense that there are constants $0<c\leq C <\infty$ such that linearization of $F$ at $u_0$ satisfies

\begin{equation} c\delta^{ij} \leq \frac{\partial F}{\partial A_{ij}}(x,u_0(x), Du_0(x), D^2 u_0(x)) \leq C\delta^{ij}\quad\text{ for all }x\in\Omega \end{equation}

in the sense of matrices. My question is whether one can conclude short time existence immediately, using uniform parabolicity at time $t=0$ and the implicit function theorem, or whether one also needs a priori estimates. I have been unable to find a reference in this level of generality, and all the papers I have read seem to gloss over this issue.

Thanks in advance!

(By the way, I am aware of the question here, although I still can't gauge from the answer there what goes into the proof in the fully nonlinear case. The specific example given is parabolic Monge-Ampere with convex initial data, but is it solely the convex initial data that yields short time existence, or the fact this also enables us to prove a priori estimates?)

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  • $\begingroup$ I think that you should assume that $F$ is an increasing function of $A$. Concavity might be important, but monotonicity is of course much more. Think to the linear case ! $\endgroup$
    – Denis Serre
    Mar 15, 2022 at 7:06
  • $\begingroup$ Whenever you have any kind of estimates for the problem as linearized around an open set of functions, it should be automatically possible to appeal to Hamilton's Nash-Moser theorem. And then the nonlinear estimates automatically follow. But this must be an overcomplication for the specific problem you are asking $\endgroup$
    – Quarto Bendir
    Mar 15, 2022 at 16:02
  • $\begingroup$ Also see Huisken and Polden's article "Geometric evolution equations for hypersurfaces" for a careful treatment of quasilinear evolution equations of arbitrary order on a manifold $\endgroup$
    – Quarto Bendir
    Mar 15, 2022 at 16:04
  • $\begingroup$ Possibly Lieberman's book "Second order parabolic differential equations" is where you want to look $\endgroup$
    – Quarto Bendir
    Mar 15, 2022 at 16:12

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Some years ago we had a similar difficulty in finding results on fully nonlinear parabolic problems. Maybe these topics fell out of fashion before anybody wrote a reasonably complete wrap-up of known results. Anyway, you might find something in our joint paper here. Actually, we consider weakly parabolic systems (i.e., the elliptic part may be degenerate) which leads to some unexpected phenomena.

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