# Short time existence for fully nonlinear parabolic equations

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

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

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 $$0 such that linearization of $$F$$ at $$u_0$$ satisfies

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

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.

• 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 ! Mar 15, 2022 at 7:06