Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.On leave from IST, Lisbon, Portugal.     

Behaviors of Solutions for a Singular Prey–Predator Model and its Shadow System

We study the asymptotic behaviors and quenching of the solutions for a two-component system of reaction–diffusion equations modeling prey–predator interactions in an insular environment. First, we give a global existence result for the solutions to the corresponding shadow system. Then, by constructing some suitable Lyapunov functionals, we characterize the asymptotic behaviors of global solutions to the shadow system. Also, we give a finite time quenching result for the shadow system. Finally, some global existence results for the original reaction–diffusion system are given.     

Asymptotic Behavior of Solutions of the Compressible Navier-Stokes Equations on the Half Space

Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space R ⁿ₊(n≥2) is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and L^p estimates for solutions of the linearized problem are obtained for 2≤p≤∞. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one that behaves like diffusion waves and the other one purely diffusively. There, however, are some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary Stokes problem in some L^p spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained. Large-time behavior of solutions of the nonlinear problem is then investigated in L^p spaces for 2≤p≤∞ by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.     

A formal finite element approach for open boundaries in transport and diffusion ground-water problems

In the application of the finite element method to diffusion and convection-dispersion equations over a ground-water domain, the Galerkin technique was used to incorporate Neumann (or second-type) and Cauchy (or third-type) boundary conditions. While mass movement through open boundaries is a priori unknown, these boundaries are usually treated as a zero Neumann condition at some far distance from the domain of interest. Nevertheless, cheaper and better solutions can be obtained if these unknown conditions are adequately incorporated in the weak formulation and in the transient solution schemes (open boundary condition). Theoretical and numerical proofs are given of the equivalences between this approach and a ‘well-posed’ problem in a semi-infinite domain with a zero Neumann condition at a boundary placed at infinity. Transport and diffusion equations were applied in one dimension to show the numerical performances and limitations of this procedure for some linear and non-linear problems. No a priori limitations are foreseen in order to find similar solutions in two or three dimensions. Thus the spatial discretization in the proximity of open boundaries could be drastically reduced to the domain of interest.     

Time and Space Fractional Diffusion in Finite Systems

Spatiotemporal nonlocal diffusion in a bounded system is addressed by considering fractional diffusion in a linear, composite system. By considering limiting conditions, solutions for combinations of Neumann and Dirichlet boundary conditions (either zero or nonzero) at the ends of a finite system are derived in terms of Mittag–Leffler functions by the Laplace transformation. Computational viability is demonstrated by inverting the solutions numerically and comparing resulting calculations with asymptotic solutions. Time and space fractional derivatives, defined by variables \(\alpha \) and \(\beta \), respectively, are employed in the Caputo sense; a single-sided, asymmetric space derivative is used. Inspection of the asymptotic solutions leads to insights on the structure of the solutions that may not be available otherwise; the resulting deductions are verified through the numerical inversions. For pure superdiffusion, characteristics of some of the solutions presented here are very similar to those of classical diffusion but combined effects for the corresponding situation result in power-law behaviors. Incidentally, to our knowledge, the pressure distribution for space fractional diffusion at long enough times in a finite system is derived based on first principles for the first time.     

Equivalent diffusion coefficient and equivalent diffusion accessible porosity of a stratified porous medium

Diffusion is an important transport process in low permeability media, which play an important role in contamination and remediation of natural environments. The calculation of equivalent diffusion parameters has however not been extensively explored. In this paper, expressions of the equivalent diffusion coefficient and the equivalent diffusion accessible porosity normal to the layering in a layered porous medium are derived based on analytical solutions of the diffusion equation. The expressions show that the equivalent diffusion coefficient changes with time. It is equal to the power average with p = −0.5 for small times and converges to the harmonic average for large times. The equivalent diffusion accessible porosity is the harmonic average of the porosities of the individual layers for all times. The expressions are verified numerically for several test cases.     

$${\mathcal{A}}$$-Stability of Global Attractors of Competition Diffusion Systems

We study the structural stability of global attractors (A{\mathcal{A}}-stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally A{\mathcal{A}}-stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C ²-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.     

Finite Propagation Speed for Limited Flux Diffusion Equations

We prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed, identified as the speed of light c. For that we construct entropy sub- and super-solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub- and super-solutions, respectively. This enables us to prove the existence of discontinuity fronts moving at light's speed.     

Attractor for a Degenerate Nonlinear Diffusion Problem with Nonlinear Boundary Condition

In this paper we prove the existence of a compact attractor in L () for a degenerate nonlinear diffusion problem with nonlinear flux on the boundary. In order to formulate the equation as a dynamical system, some existence and uniqueness results for weak solutions are proved.     

A higher-order eulerian scheme for coupled advection-diffusion transport

A new accurate high-order numerical method is presented for the coupled transport of a passive scalar (concentration) by advection and diffusion. Following the method of characteristics, the pure advection problem is first investigated. Interpolation of the concentration and its first derivative at the foot of the characteristic is carried out with a fifth-degree polynomial. The latter is constructed by using as information the concentration and its first and second derivatives at computational points on current time level t in Eulerian co-ordinates. The first derivative involved in the polynomial is transported by advection along the characteristic towards time level t + Δt in the same way as is the concentration itself. Second derivatives are obtained at the new time level t + Δt by solving a system of linear equations defined only by the concentrations and their derivatives at grid nodes, with the assumption that the third-order derivatives are continuous. The approximation of the method is of sixth order. The results are extended to coupled transport by advection and diffusion. Diffusion of the concentration takes place in parallel with advection along the characteristic. The applicability and precision of the method are demonstrated for the case of a Gaussian initial distribution of concentrations as well as for the case of a steep advancing concentration front. The results of the simulations are compared with analytical solutions and some existing methods.     

Radiative shock solutions in the equilibrium diffusion limit

A semi-analytic solution is described for planar radiative shock waves in the equilibrium diffusion (1−T) limit. The solution requires finding numerically the root of a polynomial and integrating a nonlinear ordinary differential equation. This solution may be used as a test problem to verify computer codes that use the equilibrium–diffusion radiation model, or for more advanced radiation models in the optically-thick limit. The structure of the shock profiles is also discussed, including new accurate estimates on the conditions for continuous solutions. We also discuss how the Zel’dovich spike may be estimated from the equilibrium diffusion solution. Finally, results from a computer code are shown to compare well with a semi-analytic solution.      

Dispersion in consolidated sandstone with radial flow

This paper presents some experimental and theoretical results for dispersion processes occurring in consolidated Berea sandstone with radial flow geometry. A comprehensive review of the derivation and application of several analytical solutions is also presented. The Galerkin finite element method is applied to solve the advection-dispersion equation for unidimensional radial flow.Individual and combined effects of mechanical dispersion and molecular diffusion are examined using velocity-dependent dispersion models. Comparison of simulated results with experimental data is made. The effect of flow rates is examined. The results suggest that a linear dispersion model,D=u, whereD is the dispersion coefficient,u the velocity and a constant, is not a good approximation despite its wide acceptance in the literature. The most suitable mathematical formulation is given by an empirical form of , whereD _ois the molecular diffusion coefficient. For the range of Péclet number (Pe=vd/D _m,wherev is the characteristic velocity,d the characteristic length andD _mthe molecular diffusion coefficient in porous media) examined (Pe=0.5 to 285), a power constant ofm=1.2 is obtained which agrees with the value reported by some other workers for the same regime.     

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L¹₊(R) for solutions of general nonlinear diffusion equations u_t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d₂. In the particular case of ϕ(u) ~ u^m at first order when u ~ 0, we also obtain an optimal rate of convergence in L¹ towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.     

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation as t approaches the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of the initial data.     

Existence of Traveling Waves of Invasion for Ginzburg–Landau-type Problems in Infinite Cylinders

We study a class of systems of reaction–diffusion equations in infinite cylinders which arise within the context of Ginzburg–Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non-linearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach.     

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｒｕｓｓｅｌａｔｏｒｒｅａｃｔｉｏｎｍｏｄｅｌｐｌａｙｓａｎｉｍｐｏｒｔａｎｔｒｏｌｅｂｏｔｈｉｎｂｉｏｌｏｇｙａｎｄｉｎｃｈｅｍｉｓｔｒｙ .ＳｉｎｃｅｔｈｅｍｏｄｅｌｗａｓｐｕｔｆｏｒｗａｒｄｂｙＰｒｉｇｏｇｉｎｅａｎｄＬｅｆｅｖｅｒｉｎ 1 968,ｍｕｃｈａｔｔｅｎｔｉｏｎｈａｄｂｅｅｎｐａｉｄｔｏｔｈｅｍｏｄｅｌａｎｄｍａｎｙｐｒｏｐｅｒｔｉｅｓｏｆｉｔｈａｄｂｅｅｎｒｅｓｅａｒｃｈｅｄｂｙｍａｎｙｐｅｏｐｌｅｖｉａｕｓｉｎｇｄｉｆｆｅｒｅｎｔｍｅｔｈｏｄｓ[1- 5…     

Radial Symmetry of Overdetermined Boundary-Value Problems in Exterior Domains

In this paper, we extend a classical result by J. Serrin [15] to exterior domains , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of satisfying the overdetermined boundary conditions that and u are constant on , and such that , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in [13]. The main result here covers new cases like with . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that u is constant on . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position. (Accepted May 30, 1997)     

Time-fractional radial diffusion in hollow geometries

The aim of this paper is to present the analytical solutions corresponding to the time-fractional radial diffusion in some hollow geometries. The Caputo fractional derivative is used. With the method of separation of variables and the Laplace transform, the solutions are presented in terms of the Mittag-Leffler functions. In the limitting cases, the similar solutions for the ordinary diffusion and wave equations are obtained. Furthermore, the numerical results are illustrated graphically.     

On the Chaotic Behaviour of Discontinuous Systems

We follow a functional analytic approach to study the problem of chaotic behaviour in time-perturbed discontinuous systems whose unperturbed part has a piecewise C ¹ homoclinic solution that crosses transversally the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has solutions that behave chaotically. Application of this result to quasi periodic systems are also given.