一类带有快扩散项的半导体方程的弱解存在性

In this paper, we consider the transient drift-diffusion model with fast diffusion term. This problem is not only degenerate but also singular. We present the existence result for the Neumann boundary value problem with general nonlinear diffusivities.     

Effective reaction rates of a thin catalyst layer

The catalyst layer in a fuel cell can be described with a system of reaction diffusion eq uations for the oxygen concentration and the protonic overpotential. The Tafel law gives an exponential expression for the reaction rate, and the Tafel slope is a coefficient in this law. We present a rigorous thin layer analysis for two reaction regimes. In the case of thin catalyst layers and bounded potentials, the original Tafel law enters as an effective boundary condition. Instead, in the case of large protonic overpotentials, we derive an exponential law that contains the doubled Tafel slope. Copyright © 2009 John Wiley & Sons, Ltd.     

Young measure solutions of some nonlinear mixed‐type equations

This contribution deals with measure‐valued solutions to two types of nonlinear partial differential equations for which, in general, the results on the existence of classical or weak solutions fail. These are the potential equation for transonic flow and the associated unsteady problem (forward–backward diffusion equation). The solutions are constructed by an iteration scheme (Katchanov method) and additional time discretization (Rothe method) in the second case. The existence is proved in the sense of spatial gradient Young measures. Copyright © 2010 John Wiley & Sons, Ltd.     

Dirichlet Forms Associated with Linear Diffusions

One-dimensional local Dirichlet spaces associated with linear diffusions are studied. The first result is to give a representation for any 1-dim local, irreducible and regular Diriehlet space. The second result is a necessary and sufficient condition for a Diriehlet space to be regular subspace of another Dirichlet space.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

A computational modeling of the behavior of the two-dimensional reaction–diffusion Brusselator system

This paper studies a meshfree technique for the numerical solution of the two-dimensional reaction–diffusion Brusselator system along with Dirichlet and Neumann boundary conditions. Combination of collocation method using the radial basis functions (RBFs) with first order accurate forward difference approximation is employed for obtaining meshfree solution of the problem. Different types of RBFs are used for this purpose. The method is shown to converge to the only equilibrium point of the system. Performance of the proposed method is successfully tested in terms of various error norms. In the case of non-availability of exact solution, performance of the new method is compared with the results obtained from the existing methods [7] and [8]. The elementary stability analysis is established theoretically and is also supported by numerical results.     

A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument

The paper deals with the analysis of a non-stationary parabolic partial differential equation with a time delay. The highest order derivative term is affected by the small parameter. This is precisely the case when the magnitude of the convective term becomes much larger compare to that of diffusion term. The solution of problem exhibits steep gradients in the narrow intervals of space and short interval of times. In these cases a dissipative loss turned out to be more complex. Even for the one spatial dimension and one temporal variable, not all difference scheme can capture these steep variation. Although the analysis is restricted to the model in one space dimension, the technique and comparison principles developed should prove useful in assessing the merits of numerical solution of other nonlinear model equations too.     

Global exponential stability of non-autonomous FCNNs with Dirichlet boundary conditions and reaction–diffusion terms

In this paper, we investigate the global exponential stability of non-autonomous fuzzy cellular neural networks (FCNNs) with Dirichlet boundary conditions and reaction–diffusion terms. By constructing a suitable Lyapunov functional and utilizing some inequality techniques, we obtain some sufficient conditions for the uniqueness and global exponential stability of the equilibrium solution. The result is easy to check and plays an important role in the design and applications of globally exponentially stable fuzzy neural circuits. Finally, the utility of our result is illustrated via a numerical example.     

Global asymptotic stability of positive equilibrium of three-species Lotka–Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction–diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction–diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction–diffusion system. Finally, the numerical simulation is given to illustrate our results.     

Effect of Solute‐Polymer Interactions on Sorption and Diffusion in Polyacrylamide Gels

Solute‐polymer interactions can exert a large effect on selective sorption and permeation in polyacrylamide (PAAm) gels. In order to investigate this effect, three probe polyelectrolytes, sodium polystyrene sulphonate (PSS), polyvinylpyrrolidone (PVP), and sulfonated polyaniline (SPANI), were chosen as probe species in sorption, release, and permeation experiments in PAAm gels. For PAAm gels with trapped SPANI, FTIR spectroscopy has confirmed that there exists hydrogen‐bonding between SPANI and PAAm. In addition, rigid‐chain SPANI has an intense tendency to aggregate; it is likely that the effective chains of the PAAm matrices are enwrapped in these aggregates. Hydrogen‐bonding and aggregation resulted in that the release kinetics of SPANI from PAAm gels exhibited a remarkable “lag time”, as long as 100 h (lag period means that in the initial period there is no detectable SPANI released from PAAm gels.), the releasing rate of SPANI was very slow, and the selective sorption of SPANI in PAAm gels was extremely high. On the other hand, the release and permeation of PVP and PSS through PAAm gels were much faster than SPANI, and the selective sorption were close to unity. From these facts it could be deduced that there is no or only weak interaction between PAAm and PSS (or PVP). Adding of concentrated support electrolyte resulted in decrease of the release rate and a two‐magnitude decrease of the calculated diffusion coefficients of PSS; the effect of support electrolyte on release and permeation of PSS was partly attributed to the electrostatic interaction.