In this paper, we propose a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such a system arises in semiconductors modeling and is composed of two parabolic equations and an elliptic one. We prove that the numerical solution converges to a steady state when time goes to infinity. Several numerical tests show the efficiency of the method.

We examine a cell-vertex finite volume method which is applied to a model parabolic convection-diffusion problem. By using techniques from finite element analysis, local errors away from all layers are obtained in a seminorm that is related to, but weaker than, the L2 norm.

We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.

The (almost) 1-cover lifting property of omega-limit sets is established for nonmonotone skew-product semiflows, which are comparable to uniformly stable eventually strongly monotone skew-product semiflows. These results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of ODEs, reaction-diffusion systems, differential systems with time delays ...

A Mg-ion in-diffusion process was applied to form an optical guiding structure in LiNbO(3) single-crystal fibers. A parabolic refractive-index profile was formed in a 56-microm-diameter, c-axis MgO:LiNbO(3) fiber, yielding quasi-single-mode (two modes) propagation.

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops L∞ x -norm blowup in finite time.

Using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain. We assume a polynomial type growth on the nonlinearity, and an exponential growth on the external force. The obtained results extend some existing results for non-autonomous reaction-diffusi...

In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes. 2000 Mathematics Subject Classification Numbers: 60F10, 35K55.

For a family of piecewise-autonomous one-dimensional bistable parabolic equations, with vanishing diffusion and Neumann boundary conditions, we determine the number and Morse indices of their equilibria as a function of the number of subintervals where the equations are autonomous. We conjecture how to build their attractors in a recursive way as the number of subintervals increases.

We study a two-species reaction-diffusion problem described by a system consisting of a semilinear parabolic equation and a first order ordinary differential equation, endowed with suitable conditions. We prove the existing of a unique traveling wave profile and give necessary conditions and sufficient conditions for the occurrence of penetration and conversion fronts.