相关论文: Free boundary problem governed by a non-linear dif…

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework,…

This is part II of our study on the free boundary problems with nonlocal and local diffusions. In part I, we obtained the existence, uniqueness, regularity and estimates of global solution. In part II here, we show a spreading-vanishing…

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess…

In this work we prove that the time-fractional porous medium equation on the half-line with Dirichlet boundary condition has a unique compactly supported solution. The approach we make is based on a transformation of the fractional…

We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of…

We study a toy model for the evolution of the oxygen concentration in an oxide layer. It consists in a transient convection diffusion equation in a one-dimensional domain of variable width. The motions of the boundaries are governed by the…

We study the asymptotic diffusion processes with (generally nonlocal) open boundaries in one dimension which are exactly solvable by means of the recently developed recursion formula. We investigate the stationary states, which cannot be…

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry & Lions in 2007. The results are based on a transformation of the problem to…

Fractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order…

The Kramers problem for quantum fermi-gases with specular - diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new generalised method of a source of the decision of the boundary problems…

The classical Kramers problem with specular -- diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new method of the decision of the boundary problems of the kinetic theory is stated. The…

We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ is of combustion type, coupled with the Stefan-type free boundary $h(t)$. According to [4,5], for some critical initial data, the transition solution $u$ locally…

We study the Mullins' problem that was proposed by Mullins in 1957 and is one of the models of the thermal grooving by surface diffusion. Mathematically, this is the problem of evolving curves in the half space that is governed by the…

This paper continues the investigation of Du and Lou (J. European Math Soc, to appear), where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)$ for $x$ over a varying interval…

Differential equations need boundary conditions (BC's) for their solution. It is commonly acknowledged that differential equations and BC's are representative of independent physical processes, and no correlations between them is required.…

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the steady problem for the Boltzmann equation in a general bounded domain with diffuse reflection boundary conditions corresponding to…

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu…

Exact solutions of a classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite strip bounded by the free surface (from above) and (from the…

We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…

In this paper, we study a linear convection-diffusion equation with time-dependent coefficients on a bounded interval. The problem includes inhomogeneous Dirichlet boundary conditions and is motivated by physical models where the…