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.     

Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).     

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...     

Multi-Vortex Solutions to Ginzburg–Landau Equations with External Potential

We consider the existence of multi-vortex solutions to the Ginzburg–Landau equations with external potential on \mathbbR²{\mathbb{R}^2} . These equations model equilibrium states of superconductors and stationary states of the U(1) Higgs model of particle physics. In the former case, the external potential models impurities and defects. We show that if the external potential is small enough and the magnetic vortices are widely spaced, then one can pin one or an arbitrary number of vortices in the vicinity of a critical point of the potential. In addition, one can pin an arbitrary number of vortices near infinity if the potential is radially symmetric and of an algebraic order near infinity.     

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 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.     

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.     

Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN

This paper is devoted to time-global solutions of the Fisher-KPP equation in ℝ ^N :

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...     

The Continuous Coagulation-Fragmentation¶Equations with Diffusion

Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L ¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.     

On Nonexistence for Stationary Solutions to the Navier–Stokes Equations with a Linear Strain

We consider stationary solutions to the three-dimensional Navier–Stokes equations for viscous incompressible flows in the presence of a linear strain. For certain class of strains we prove a Liouville type theorem under suitable decay conditions on vorticity fields.     

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.     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

Unique Solutions to Hartree–Fock Equations for Closed Shell Atoms

In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree–Fock equations of atoms. We show, for example, that the Hartree–Fock ground state of a closed shell atom is unique provided the atomic number Z is sufficiently large compared to the number N of electrons. More specifically, a two-electron atom with atomic number Z\geqq 35{Z\geqq 35} has a unique Hartree–Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some Z > 1, which exhibits a phase segregation.     

Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions

We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov–Vicsek models that can be considered as non-local, non-linear, Fokker–Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin–Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339–343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193–1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov–Vicsek models is the unit sphere. Our analysis for L^p estimates and compactness properties take advantage of the orientational interaction property, meaning that the velocity space is a compact manifold.