# Existence and Uniqueness of Solutions to Quasilinear Parabolic PDEs

Consider the following general form of a quasilinear parabolic PDE $$u_t = a(x,u,u_x)u_{xx} + b(x,u,u_x) \ \ \textrm{ for }-1<x<1, \tag{1}$$ with inhomogeneous boundary condition $u_x(x,t) = g(u(x,t))$ for $x=\pm 1$.

I am looking for references that shows the following: for any initial data $u_0$ living in some suitable Holder space, there exists a unique local-in-time classical solution $u$ in some time interval $0\le t<T$, in the sense that $u,u_x$ are continuous in $[-1,1]\times [0,T)$, $u_{xx}, u_t$ are continuous in $(-1,1)\times (0,T)$ and $u$ satisfies equation (1) in $(-1,1)\times (0,T)$.

I knew of a paper by Lunardi: "Abstract Quasilinear Parabolic Equations" (See here https://doi.org/10.1007/BF01456097) that proves the claim by establishing some a-priori gradient estimate. Correct me if I am wrong, but it seems to me that it only works for zero Neumann condition. If it helps, the equation I had in mind actually stems from curvature driven flow in a two-dimensional domain.

• I'm not sure I understand the boundary condition - is there a typo? – Anthony Carapetis Oct 26 '17 at 3:36
• @AnthonyCarapetis It should be $u(x,t)$, unless you meant otherwise? – Chee Han Oct 26 '17 at 4:14
• Should it really be $g(u(x,t))$ and not just $g(x,t)$? As written it's some kind of weird fixed-point condition that is certainly not well-posed for all $g$ (e.g. $g = \mathrm id$ makes the boundary condition vacuous). – Anthony Carapetis Oct 26 '17 at 4:17
• @AnthonyCarapetis You are absolutely right. It was supposed to be Neumann instead of Dirichlet boundary conditions. I have edited the question. – Chee Han Oct 26 '17 at 5:25
• OK, that makes more sense. I believe that with the right assumptions on $g$, you should be able to reference chapter 8 of Lieberman for the existence (see the section on oblique derivative problems) and chapter 9 for the uniqueness. – Anthony Carapetis Oct 26 '17 at 5:32