Geometric theory of one-dimensional nonlinear parabolic equations. I. Singular interfaces

Abstract

We present the concepts of a geometric analysis of the free-boundary problems for a nonlinear one-dimensional parabolic PDE with strong singularities $$ u_t = F(u,u_x,u_{xx}), \quad x \in {{\bf R}}, \,\, t>0; \quad u(x,0) = u_0(x) \ge 0, \quad x \in {{\bf R}}, $$ where the function $F(p,q,r)$ is smooth for $p>0$, satisfies the parabolicity condition $F_r(p,q,r) > 0$ for $p>0$ and, in general, is not defined (singular) at $p=0$. A proper maximal solution $u(x,t) \ge 0$ is the limit of a monotone decreasing sequence of smooth strictly positive solutions of the regularized problems, which are assumed to be well-posed. The solutions exhibit finite interface propagation on the singular level $\{u=0\}$. The geometric theory reduces the study of the PDE to a family of the ODEs. We show that existence, Bernstein-type gradient estimates, moduli of continuity, interface regularity, the interface equation, etc. are directly connected with the family of ODEs associated with the nonlinear PDE. For the autonomous PDEs we utilize a complete set $B=\{V\}$ of particular travelling wave solutions $V(x,t)=f(x-{\lambda} t)$, where $f(\xi)$ solves a nonlinear ODE with a parameter ${\lambda} \in {{\bf R}}$. The proofs rely on intersection comparison techniques of the solutions $u(x,t)$ with the family $B$ which are based on the classical Sturm Theorem on zero sets for linear parabolic equations. We study the regularity properties and derive the interface equations for several types of the quasilinear and fully nonlinear equations from combustion, filtration and detonation theory.