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.

We study the large time asymptotic behavior of solutions to the periodic problem for a model nonlinear evolution equation. This equation has a general structure, it contains many well-known equations of mathematical physics, such as: the Korteweg-de Vries equation and nonlinear Schrödinger equation. We find the asymptotic representation of solution. Depending on the linear part of equation and the structure of the nonlinearity the solution can exponentially decay with time, oscillate or grow exponentially with time. Taking into account the symmetry of the nonlinear term we consider the case of large initial data.

Let $V=V(\Omega)$ be a set satisfying the Poincaré-Sobolev pointwise inequalities for the relative rearrangement and $\rho$ an arbitrary norm on the set of measurable functions on the interval $\Omega_*=(0,measure(\Omega))$. Then using the associate norm of $\rho$, we give some sufficient conditions to ensure that a function $u$ of $V$ has to be bounded or integrable in an Orlicz space. We show similar results for a solution of quasilinear equation under some conditions between the data and $\rho$. We then give an unified approach for many kinds of estimates. Using a main relation involving some pointwise relations for the relative rearrangement, we prove a regularity result for the time derivative of a parabolic system well posed in a Grand Sobolev space.