We tackle the well-posedness of certain dynamical systems that result in non-autonomous quasi-linear problems in a critical setting, where the coefficients defining the flux and the Neumann boundary conditions depend on the solution itself. We want to show the existence and uniqueness of these solutions on a very short timescale.

The local well-posedness of quasi-linear problems in a critical setting by the maximal $L^p$-regularity theory from Chapter 18 of Hytönen, van Neerven, Veraar, and Weis (2024) are investigated, where we use the non-autonomous setting with non-constant domains from Di Giorgio, Lunardi, and Schnaubelt (2005). Dominant examples in the literature of such problems are problems with multidimensional, non-constant Neumann boundary conditions influencing the domain of the operator. In this thesis, we look for ways to ensure the short-timescale existence and uniqueness of solutions to these problems and research the possibility of applying them to the model problem with Neumann boundary conditions. By applying non-autonomous linear theory from Chapter 3 Part II of Yagi (2010), we find a result that allows us to determine the existence and uniqueness of short timescale mild solutions. When applied, however, we see that, unlike the work of Yagi (2010), we can only guarantee the local well-posedness of the model Neumann problem by using averaging functions because of our more strict critical setting. In the future, results can be based on different regularity types, or the given result can be applied on spaces with negative smoothness.

When transforming PDE problems using Fourier and Laplace transforms, we can find functions that represent the problem, and which can be used to determine properties of the problem. We define such functions as symbols $P(\lambda,z)$. In general, we define the class of symbols $S(L_t\times L_x)$ are all functions which are represented by a polynomial of the form $R_P(\lambda,z):=\sum_{\ell\in I_P}\tau_\ell(\lambda,z)\phi_\ell(\lambda)\psi_\ell(z)$, where $\tau_\ell(\lambda,z)$ are $\rho$-homogeneous functions of $(\lambda,z)$ on the cones $L_t\times L_x$, and $\phi_\ell(\lambda)$ and $\psi_\ell(z)$ homogeneous functions of $\lambda$ on the cone $L_t$ and $z$ on the cone $L_x$ respectively. These functions have a certain $\gamma$-order $d_\gamma(P)$ that shows the order of the function relative to a relative weight $\gamma$, and a certain $\gamma$-principal part $\pi_\gamma P(\lambda,z)$, which is the part of $P$ that causes this $\gamma$-order.

For such a symbol, we define its Newton polygon $N(P)$ as a certain convex hull of points on $[0,\infty)^2$, which serves as a geometric description of the order of $P$. We define the weight function $W_P$ to be a positive polynomial with the orders found on the vertices of the Newton polygon $N(P)$. We define a notion of order functions, and show the order function of $P$ is $d_\gamma(P)$.

We define a notion of parameter-ellipticity and parabolicity for symbols in $S(L_t\times L_x)$ based on the Newton polygon $N(P)$, namely N-parameter-ellipticity and N-parabolicity. Using various results from the work of R. Denk and M. Kaip \cite{Denk Kaip} and results I introduced myself, we then prove the equivalence between N-parameter-ellipticity and having non-vanishing $\gamma$-principal parts.