Uniform Convergence for Approximate Traveling Waves in Linear Reaction–Diffusion–Hyperbolic Systems

In this paper we study linear reaction–hyperbolic systems of the form , (i = 1, 2, ..., n) for x > 0, t > 0 coupled to a diffusion equation for p ₀ = p ₀(x, y, θ, t) with “near-equilibrium” initial and boundary data. This problem arises in a model of transport of neurofilaments in axons. The matrix (k _ij ) is assumed to have a unique null vector with positive components summed to 1 and the v _j are arbitrary velocities such that . We prove that as the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is , for any small positive α.     

Random Dynamics of Stochastic Reaction–Diffusion Systems with Additive Noise

For a typical autocatalytic stochastic reaction–diffusion system with additive noises, the multicomponent reversible Gray–Scott reaction–diffusion system on a two-dimensional bounded domain, the existence of a random attractor and its attracting regularity are proved through the sharp uniform estimates showing respectively the pullback absorbing, asymptotically compact, and flattening properties.     

Global Existence of Renormalized Solutions to Entropy-Dissipating Reaction–Diffusion Systems

Archive for Rational Mechanics and Analysis - In the present work we introduce the notion of a renormalized solution for reaction–diffusion systems with entropy-dissipating reactions. We...     

Complex Attractors and Patterns in Reaction–Diffusion Systems

We consider semiflows generated by initial boundary value problems for reaction–diffusion systems. In these systems, reaction terms satisfy general conditions, which admit a transparent chemical interpretation. It is shown that the semiflows generated by these initial boundary value problems exhibit a complicated large time behavior. Any structurally stable finite dimensional dynamics (up to an orbital topological equivalence) can be realized by these semiflows by a choice of appropriate external sources and diffusion coefficients (nonlinear terms are fixed). Results can be applied to the morphogenesis and pattern formation problems.     

On Dirichlet Problem for a Class of Delayed Reaction–Diffusion Equations with Spatial Non-locality

We consider a very general class of delayed reaction–diffusion equations in which the reaction term can be non-monotone as well as spatially non-local. By employing comparison technique and a dynamical system approach, we study the global asymptotic behavior of solutions to the equation subject to the homogeneous Dirichlet condition. Established are threshold results and global attractiveness of the trivial steady state, as well as the existence, uniqueness and global attractiveness of a positive steady state solution to the problem. As illustrations, we apply our main results to the local delayed diffusive Mackey–Glass equation and the nonlocal delayed diffusive Nicholson blowfly equation, leading to some very sharp results for these two particular models.     

On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R $ , and $s\ge 1$ , $s\ge r$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.     

On the Time Decay of the Solutions of the Navier–Stokes System

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator. This work was done while the second author was visiting the Bernoulli Center, EPFL, Switzerland, whose support is gratefully acknowledged.     

Exponential Propagation for Fractional Reaction–Diffusion Cooperative Systems with Fast Decaying Initial Conditions

We study the time asymptotic propagation of solutions to the reaction–diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.     

On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate \({O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}\) , are exactly characterized by their initial conditions belonging into the homogeneous Besov space \({\dot{B}^{-\alpha}_{2, \infty}}\) . Similarly, for a solution u and \({p \in [1, \infty]}\) the integral \({\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}\) is finite if and only if the initial condition of u belongs to the homogeneous Besov space \({\dot{B}_{2, p}^{-\alpha}}\) . For the case \({\alpha \in (5/2, 9/2]}\) we present analogical results for some subclasses of turbulent solutions.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy

Hyperbolic–parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier–Stokes system.     

Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

Journal of Dynamics and Differential Equations - We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and...     

Spatiotemporal Patterns of a Reaction–Diffusion Substrate–Inhibition Seelig Model

In this paper, the spatiotemporal patterns of a reaction–diffusion substrate–inhibition chemical Seelig model are considered. We first prove that this parabolic Seelig model has an invariant rectangle in the phase plane which attracts all the solutions of the model regardless of the initial values. Then, we consider the long time behaviors of the solutions in the invariant rectangle. In particular, we prove that, under suitable “lumped parameter assumption” conditions, these solutions either converge exponentially to the unique positive constant steady states or to the spatially homogeneous periodic solutions. Finally, we study the existence and non-existence of Turing patterns. To find parameter ranges where system does not exhibit Turing patterns, we use the properties of non-constant steady states, including obtaining several useful estimates. To seek the parameter ranges where system possesses Turing patterns, we use the techniques of global bifurcation theory. These two different parameter ranges are distinguished in a delicate bifurcation diagram. Moreover, numerical experiments are also presented to support and strengthen our analytical analysis.     

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.     

On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.     

Entire Solution in Cylinder-Like Domains for a Bistable Reaction–Diffusion Equation

We construct nontrivial entire solutions for a bistable reaction–diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki et al. (Calc Var Partial Differ Equ 55(3):1–32, 2016) concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the “cylinder-like” assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We need a different approach based on the long time stability of the bistable wave in heterogeneous media. We also prove the existence of an entire solution for a one-dimensional problem with a non-homogeneous linear term.     

On Spatial Decay Estimates for Derivatives of Vorticities with Application to Large Time Behavior of the Two-Dimensional Navier–Stokes Flow

In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the Oseen vortex.      

Existence and Characterization of Attractors for a Nonlocal Reaction–Diffusion Equation with an Energy Functional

In this paper we study a nonlocal reaction–diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.

     

Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

On the Shape of Meissner Solutions to a Limiting Form of Ginzburg–Landau Systems

In this paper we study a semilinear system involving the curl operator, which is a limiting form of the Ginzburg–Landau model for superconductors in \({{\mathbb{R}}^3}\) for a large value of the Ginzburg–Landau parameter. We consider the locations of the maximum points of the magnitude of solutions, which are associated with the nucleation of instability of the Meissner state for superconductors when the applied magnetic field is increased in the transition between the Meissner state and the vortex state. For small penetration depth, we prove that the location is not only determined by the tangential component of the applied magnetic field, but also by the normal curvatures of the boundary in some directions. This improves the result obtained by Bates and Pan in Commun. Math. Phys. 276, 571–610 (2007). We also show that the solutions decay exponentially in the normal direction away from the boundary if the penetration depth is small.